I don't have a specific answer to your question, but I did find a particularly interesting section in Andrew Chaikin's 1994 book, A Man on the Moon. In a way, this particular excerpt explores the sort of gap between people's spirituality and scientific understanding that became apparent following the moon landings:

In 1975, during a trip to Nepal with his wife, Joan, [Apollo 14 Command Module Pilot] Stu Roosa visited a school to give a talk about his flight to the moon. Afterward, there were mysterious questions: "Who did you see?"

Roosa answered, "There is no one there." A murmur went through the place. Again the students asked what he had seen. Roosa was adamant: "There is nothing there. Not even wind. There is nothing."

Later, after the Roosas had gone, a teacher told the children, "You mustn't listen to him. He's wrong."

The Roosas were distressed when they learned that some Nepalese believe the spirits of their ancestors reside on the moon. Roosa had essentially told them there was no heaven. Joan wished the American government had briefed them better for the trip.

One day at the hotel, Joan went to get her hair done. The woman giving her a shampoo said, "You're married to the astronaut, aren't you?"

"Yes, I am."

"You know, you're married to a god." Joan laughed it off. But when she and Stuart went to that evening's gathering, all along the mountain road children were kneeling with candles, in an act of reverence: a god had come to visit. (Chaikin, 553-4)

Chaikin, Andrew. A Man on the Moon. New York: Penguin Books, 1994. Print.

Edit:

I can't speak in terms of religious opposition, but I did recall reading of it being the other way around.

On Christmas Eve (December 24), 1968 as Apollo 8 was in orbit around the Moon, crew members Bill Anders, Jim Lovell, and Frank Borman took turns reciting verses 1 through 10 of Genesis, using the King James Version text. You can listen to the recording here.

Following the reading, the founder of American Atheists, Madalyn Murray O'Hair, sued the United States government, alleging violations of the First Amendment.

The Supreme Court responded by dismissing the suit due to a lack of jurisdiction.

On Apollo 11 after landing, Buzz Aldrin became the first individual to ever received communion on the lunar surface. Due to the response to the Genesis reading, Buzz abstained from reading scripture on-air, instead doing so to himself, though he was recorded as saying on-air:

"This is the LM pilot. I'd like to take this opportunity to ask every person listening in, whoever and wherever they may be, to pause for a moment and contemplate the events of the past few hours and to give thanks in his or her own way."

There was some significant protest by the Southern Christian Leadership Council - A collection of black church leaders.

200 Black Protestors marched against the Apollo 14 mission (this is of course after we'd already once landed on the moon) - See Rome News-Tribune Feb 2, 1971.

But this was more of a Socio-Economic Protest than a truly religious one, as the main quote from the Article quotes one member of the council stating that it's just a bunch of lazy white boys up there looking for rocks and if they had wanted real work to be done they'd sent a n-----.

What's more stunning is the scientific community's opinion on the Space Program. A 1958 report to the President from his Science Advisory Committee included commentary that Sociologist Amitai Etizioni paraphrased as "some of the most eminent scientists in this country" bagged on our space ambitions.

"Research in outer space affords new opportunities in science but does not diminish the importance of science on earth,"

And concludes with, "It would not be in the national interest to exploit space science at the cost of weakening our efforts in other scientific endeavors. This need not happen if we plan our national program for space science and technology as part of a balanced effort in all science and technology."

In fact, he felt so strongly against the space program, he wrote a book called Moondoggle, which is the basis for the majority of the article I'm quoting.

His book also includes a Science poll of 113 scientists not associated with NASA found that all but 3 of them "believed that the present lunar program is rushing the manned stage. Etzioni's final assessment -- "most scientists agree that from the viewpoint of science there is no reason to rush a man to the moon"

Finally, there's the fact that Buzz Aldrin, at the time was a local leader in his church and performed Communion Quietly onboard the Eagle Landing Module - Communion In Space by Buzz Aldrin, Guideposts, October, 1971

Of course this is just from a cursory search, and I'm sure actual experts in this field of History might have access to more obscure texts that could prove Religious Objection, but it seems that for the most part, it wasn't seen as an affront to religion to explore space.

Edit* I forgot to include the article I was pulling a lot of those quotes from It was from the Atlantic

Mormons have a long history of moon problems. In 1961, Joseph F. Smith, then president of the quorum, later president of the church, said:

"We will never get a man into space. This earth is man's sphere and it was never intended that he should get away from it. The moon is a superior planet to the earth and it was never intended that man should go there. You can write it down in your books that this will never happen."

He was asked about this statement again in 1970 and was good enough to answer "Well, I was wrong, wasn't I?".

It's important to say that he wasn't the prophet (who supposedly has a direct link to God) when he originally made the statement. From the wikipedia article on the prophet:

"Latter-day Saints consider the president of the church to be God's spokesman to the entire world and the highest priesthood authority on earth, with the exclusive right to receive revelations from God on behalf of the entire church or the entire world."

If only they'd made him prophet earlier, then God could have told him it was possible.

http://en.wikipedia.org/wiki/Joseph_Fielding_Smith

I'd like to add a question to your question if that's alright OP: Did people protest the launch of the rocket and moon landing because of fear of "moon men" or some astrological signs? Was that a common thing in the 60s?

May I add on to your question? I want to know how people from other nations reacted when they put the American flag up there. Like for example, did the Soviet react strongly against it? Or was everyone okay with it because it was a US mission?

I can't find any evidence of organized opposition on religious grounds and I've searched a few academic databases now.

There is a bit of controversy over religious references made on the moon and during the mission.

NASA asked Neil Armstrong and Buzz Aldrin to not make any overt religious references (both were religious, Methodist and Presbyterian) while on the moon and they respected it. They did take Communion on the moon though not publicly. They choose to keep this hidden after a lawsuit about them reading Genesis during the mission.

"Perhaps, if I had it to do over again, I would not choose to celebrate communion. Although it was a deeply meaningful experience for me, it was a Christian sacrament, and we had come to the moon in the name of all mankind – be they Christians, Jews, Muslims, animists, agnostics, or atheists. But at the time I could think of no better way to acknowledge the enormity of the Apollo 11 experience than by giving thanks to God."

They seem to have found a happy medium to a degree. Undergoing personal religious experiences while publically not allowing them to dominate the conversation. A bit of the Streisand effect happened with the lawsuit by an atheist activist over the reading of Genesis. No one had really cared until the lawsuit happened. Then suddenly it had religious significance.

http://www.theguardian.com/commentisfree/belief/2012/sep/13/buzz-aldrin-communion-moon

As a counterpoint, Buzz Aldrin himself conducted communion on the moon. It was even the first food consumed on the moon.

Source: Magnificent Desolation: The Long Journey Home from the Moon by Buzz Aldrin

This is perhaps just a bit earlier than the time period you are looking (predating the moon landing by about two decades), but I recently read a book by James Gilbert called Redeeming Culture: American Religion in an Age of Science. One of his overarching points is that around the time of Roswell and the Yakima incident, when space and UFO's captured the popular American imagination, there was a kind of cultural struggle to find the terms to talk about this space "beyond" what we can see.

One of the chapters is dedicated to discussing the life of Wernher von Braun, "the central figure in the Nazis' rocket development program... taken to the United States as part of the then-secret Operation Paperclip." von Braun is quoted in Redeeming Culture as saying:

"'Since we first began the exploration of space through rocketry, we have regularly received letters expressing concern over what our writers call 'tampering' with God’s creation.' One letter, he noted, even “revealed an honest fear that a rocket would strike an angel in space high above the earth."

In response, Gilbert writes, von Braun was faced with the task of demonstrating the compatibility of religious and scientific discourses to describe space and the beyond. In fact, I believe Gilbert makes the argument that doing so was a condition for the funding and popular support necessary for his research to continue.

The book is a fascinating read and I would recommend it. I'm not sure how directly it addresses your original point but I thought it was a fascinating anecdote.

In regard to religious opposition, I have a memory from when I was about 9, but I can't find anything to support it. Jack Wyrtzen, a well known evangelist, came to the Altoona Bible Church (PA) during a missionary conference and spoke on JFK's intention to put a man on the moon. I remember him using words like trajectory and escape velocity which were beyond my years, but I understood what he was talking about. He said it wasn't going to happen. God never intended it and it was not technically possible. Then there was the opposition to a Catholic president that was prevalent at the time. It was decades ago but I've had that memory ever since, punctuated by the fact that it happened. There is a fair amount of info on Wyrtzen but I'd be surprised to find this and he may not have published it. There must have been many people who heard it though.

/u/sinisterstarr made a comment regarding a leader of the Church of Jesus Christ of Latter-day Saints (better known as "Mormons"), Joseph Fielding Smith. During the 1960s, he said more than once that God would never allow man to land on the moon. At that time, he was President of the Quorum of the Twelve Apostles of the church. He became President of the Church in January 1970, between the Apollo 12 and Apollo 13 missions.

I'm posting my responses here as a main comment on the suggestion of /u/HumanPlus. Enjoy!

IIRC, the Apollo 12 astronauts took a Utah state flag to the moon, then autographed it with Sharpies or whatever, and presented it to President Smith. I was very amused to hear that.

Sorry, it was Apollo 15. Deseret News, Sept. 18, 1971.

Also, here's a page with good sources on what President Smith said and how and when.

Here is a song about it.