My impression is that in the ancient world and in some cultures infanticide was and is considered more a form of "family planning" than a kind of murder.
I'm wondering if there were any tensions in the Roman Empire during the spread of Christianity similar to the modern day abortion debate.
Well, I'll start with the early churches thoughts on the subject. Those books that were canonized in the New Testament do not mention abortion. The closest the Old Testament comes is perhaps Exodus 21:22-25 that discusses a man being punished for killing/harming an unborn child by hitting a woman.
However, other early Christian writings do discuss the topic. The Didache is a major one. The Didache, also known as The Teachings of the Twelve Apostles, is a work that most believe was written sometime in the late first century or early second century. It's kind of a summary of everything you'll read in the New Testament, though it does branch out and talk about the details of how the early church was practicing baptism and some other topics that aren't so fleshed out in the NT. Here is the passage that deals with abortion:
"This is the second commandment of the teaching: do not murder, do not commit adultery, do not corrupt boys, do not be sexually promiscuous, do not steal, do not practice magic, do not murder a child by abortion or kill a newborn infant..."
The Apostolic Father Tertullian had plenty to say on both the subject of abortion and when, exactly life begins.
"In our case, murder being once for all forbidden, we may not destroy even the foetus in the womb, while as yet the human being derives blood from other parts of the body for its sustenance. To hinder a birth is merely a speedier man - killing; nor does it matter whether you take away a life that is born, or destroy one that is coming to the birth. That is a man which is going to be one; you have the fruit already in the seed."
And again,
"That the unborn child is alive:
How are they dead unless they were first alive? But still in the womb an infant by necessary cruelty is killed when lying twisted at the womb's mouth he prevents birth and is a matricide unless he dies. Therefore there is among the arms of physicians an instrument by which with a rotary movement the genital parts are first opened, then with a cervical instrument the interior members are slaughtered with careful judgment by a blunt barb, so that the whole criminal deed is extracted with a violent delivery. There is also the bronze needle by which the throat - cutting is carried out by a robbery in the dark; this instrument is called and embryo knife from its function of infanticide, as it is deadly for the living infant."
So, as to your specific question. The Romans used a couple types of abortion methods, as you can see above. But also they used an herb called silphium, which is now extinct thanks to the Greeks, but the seed of that plant might be where we got our heart symbol from.
We know at least some Romans were against the practice of abortion because of the Hippocratic oath (Do a quick google of Scribonius Largus), but others argued for Aristotle's view of delayed ensoulment. In particular Soranus writes of two different parties among physicians. Christianity spread very quickly, however, and by C.E. 211 Septimus Severus banned abortion completely.
As far as evidence of tensions, I can't say that I know of any, or have been able to locate any. My guess would be that there were personal issues about it, but I am unaware of any political tensions on the subject.
Allen, George Cantrell. The Didache: or, The Teaching of the Twelve Apostles. London: Astolat Press, 1903.
Tertullian, Henry BROWN, and Thomas WESTLEY. Tertullian's Apology, or defence of the Christians, against the accusations of the Gentiles. Now made English by H(enry) B(rown). London, 1655.
"Soranus' Gynecology." Google Books. Accessed January 05, 2017. https://books.google.com/books?id=YsKWfh31gxwC&lpg=PR1&dq=soranus gynecology digital&pg=PR1#v=onepage&q&f=false
Ancient Greek society also applied some sort of fitness test upon the newly born. It was applied in Ancient Sparta by Spartiate fathers to ensure the strength of the collective and as much as possible ensure that weakness and deformity were not handed down through the generations. The story 'Oedipus Rex' clearly shows it to be common practice in Greece, at least in the BCE era.