The short answer would be no. Although sons might have been viewed as more important as they carried on the family name, could serve in the military and hold important offices in both politics and religion I could find no information on female babies being killed simply for being born first. In the book "Ancient Rome: A History" by D.Brendan Nagle it does mention infanticide as the most common form of population control in the ancient world but does not provide any statistics nor say one gender was killed more than the other. It actually has a quote by the Greek statesmen Polybius, writing in the 2 B.C.E. Rome, that this practice is a threat to both the state and individual households. "A Brief History of the Romans" by Mary T. Boatwright, Daaniel J. Gargola, Richard J.A.Talbert also seems to support this idea as it briefly mentions children when discussing marriage and in that section reveals that 3/5 children would not make it to adulthood. So simply killing a healthy child for being a girl seems like an unlikely scenario. That being said it is always possible that this did happen in some cases, although it would probably be an exception and not the rule.

It happened, but whether it was "common" is not agreed upon (and depends on what is meant by "common"). Infanticide of children is hardly restricted to Ancient Rome, and is well documented, but how frequently females were targeted is less known.

Ancient Romans, and many other cultures throughout history, practiced both infanticide and "exposure." Exposure can lead to, but is not identical to, infanticide. It's when an unwanted baby was abandoned, to fend for itself. Usually this happened immediately after birth, as keeping the baby for even a few days meant that the baby was accepted by the father. Exposure does not always mean that the baby would die. In some cases the midwife would know of a family wanting another child and hand the baby off. Or a baby could be left on the side of the road and picked up by a stranger wanting a child. Etc. I don't know how often exposure lead to death (vs adoption), but it seems implied from what I've read that death was far more common. Harlow and Lawrence say, "Exposed children were often raised by others,"^1 but give no explanation or statistics to back that up.

There are many reasons for practicing exposure and infanticide, as there are many reasons for an unwanted pregnancy: no father/child born out of wedlock, unplanned birth when the mother needs to work, family can't support another mouth to feed, recent financial hardship, recent divorce/death of father, child born from an adulterous affair or from incest, child was born deformed, etc.

There are also reasons where the sex of the child was a factor in its being unwanted: a father not wanting to divide up his son's inheritance with another son, needing a son to do hard labor, not wanting to pay for a daughter's dowry, etc. Those last two together tip the scales for why sons would be preferred to daughters, in many cultures with dowry payments.

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So there were reasons that the sex of the child mattered, but did that show in practice?

Most of what I've read has said yes or probably:

"A study of ethnographical reports on 393 widely scattered cultures shows that 179 commonly and 29 occasionally practised infanticide, predominantly of girls."^2 [These are reports from all over time and place, not just Ancient Rome.]

"Wherever infanticide is practised, female infanticide is the rule... supplemented by the elimination of defective and unhealthy offspring and those undesirable by reason of some magical (e.g., multiple births) or social (e.g., illegitimacy) factor. Infanticide is thus associated with the higher valuation of males."

But The Oxford Handbook of Childhood and Education in the Classical World says that it was a factor, but:

...the case for widespread exposure of females has been hugely overblown, fueled by a few often repeated pieces of evidence and a belief that in the ancient world sons were preferred to daughters and therefore that female infants were less likely to be raised (see Scott 1999: 67-73, 2001a: 144-6, 2001b). This assumes not only that a preference for one sex will result in the casting out of the other but also that a prejudice expressed only sporadically (and usually in literary contexts) was applied universally in the Greek and Roman World.

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1. Mary Harlow and Ray Lawrence, Growing Up and Growing Old in Ancient Rome. p39. 2005.

2. Mark Golden, Demography and the Exposure of Girls at Athens. 1981. http://www.jstor.org/stable/1087926

3. Editors: Judith Evans Grubbs, Tim Parkin, Roslynne Bell, The Oxford Handbook of Childhood and Education in the Classical World. p90. 2013.