In the Church of England itself, no, or at least if there were such objections among the moderate Protestant wing of the church, they didn't get very far.

But among the Puritan movement in the Anglican Church, yes. This was one of the many gripes that the Puritans had with the Protestants. The Puritans felt the Protestants were keeping the Church too "papist" or "romish" in the 16th and 17th Centuries, as they would have said. There was too much pagan-based, non-scriptural superstition in the Catholic church, and the King's Protestant allies were not purifying the church of these superstitions nearly enough for the Puritans' liking.

As is pointed out in Stephen Nissenbaum's The Battle For Christmas and in Charles E. Hambrick-Stowe's The Practice of Piety: Puritan Devotional Disciplines in Seventeenth-Century New England, among other places, surviving calendars and almanacs from 17th Century New England tend to reject the Monday/Tuesday/etc. naming convention. Hambrick-Stowe's book quotes from the 1690 Poor Man's Pocket-Book: Being a Perpetual Almanack of Spiritual Meditations, which contained a "Compleat Directory of one endless Week":

"Childhood for Monday, Youth for Tuesday, Manhood for Wednesday, Old-age for Thursday, Death for Friday, Judgement for Saturday, Eternity for Sunday..."

However, as Nissenbaum points out, this trend against pagan days of the week was rather short-lived, while the rejection of holidays that the Puritans judged as being pagan-based (such as Christmas and New Year's) continued much longer.

The plainer tradition of the Quakers took the rejection of the pagan names in the calendar even further, and it lasted much longer. "The Quaker Calendar" by Samuel G. Barton, published in Proceedings of the American Philosophical Society, gives a rundown of the history, and the naming conventions are also recounted in The Quaker Family in Colonial America by Jerry G. Frost, and in Albion's Seed: Four British Folkways in America by David Hackett Fischer. The Quakers not only rejected the days of the week, but also the months of the year. Sunday became "First Day", Monday was "Second Day", and so on. The months were similarly denoted, with March being the "First Month", April the "Second Month" and so on, based upon the Bible passage Exodus 12:2 which calls the month of Passover the first month:

"This month shall be unto you the beginning of months: it shall be the first month of the year to you."

With the adoption of the Gregorian calendar by Parliament in 1752, the month names shifted, with January becoming the "First Month".

There were several schisms in the Quaker tradition during the 19th Century, so while these conventions did survive at least among some Quakers into the latter half of the 1800s, I am not sure if there are still any Quaker groups that use them. But at the very least, Quakers still tend to call their version of Sunday School "First Day School" to the present day.

EDIT: Clarity.