I think it's hard to say definitively. For one, birth control and abortifacients have been around long before the modern pill or forms of abortion, although they could be dangerous or less successful.

One interesting case of infanticide can be found in the Marshall Islands. The pre colonial Marshallese did not- or I have not read about- have a form of contraception, although abortion certainly did exist, both physical and medical in nature. However, early reports by 19th century colonialists and travellers report infanticide as having occurred.

To give a bit of background on family life: in pre colonial Marshallese society, there were multiple clan systems and also a kinship system, which was essentially an extended family, known as a bwij, overseen by an alap, and paying tribute and ruled by an iroij/chief.

In the 19th century, girls tended to marry between 12-14, boys a few years later, so at the beginning or during the start of puberty. Although sexual intercourse was expected after puberty, having children was not. Sex was not purely or even largely for the purpose of procreation (although that doesn't mean their sex wasn't penetrative, I don't think there's much information on the sexual habits of the pre colonial Marshallese).

There were many rules surrounding sexual etiquette, and in some ways these regulations did prevent having too many children. There were a few sexual taboos, the first being incest, that is, sex between members of the same bwij, not necessarily direct relatives. When menstruating, women did not have sex or contact. After childbirth, women were not allowed to have sex for some time. After the death of an iroij or a disaster, those involved in the funeral did not have sex for some time. When ill or undergoing some form of treatment, sex was prohibited. Women had less access to members outside their bwij. As mentioned above, abortion was available, but mostly used by newlyweds or unmarried girls.

So with no contraception, but children limited through sexual taboos and to a minor extent, abortion, there was another possible solution to the fact that having more than a couple of children while on a small atoll without mass or stable agriculture and with unreliable fish and very little trade beyond their chain of atolls, was dangerous for the survival of the community: infanticide.

It's kind of a thorny issue, actually. Did it happen? Almost all early reports say that yes, infanticide certainly did happen, usually by setting babies on small rafts and allowing them to drown out at sea, with a minority of reports involving burying alive or some form of burial ceremony. To what extent this actually happened is the issue. In the Ratak Chain, it was traditionally reported as having occurred routinely: any child born after the second killed in some reports, every child born after the fourth in more, albeit only in the Ratak Chain. However Marshallese oral history and a variety of other reports dispute this, tending to agree on the topic of infanticide, and it is on this side of the line that I also fall: although systematic infanticide may have occurred at some point during pre-colonial history, by the early 19th century, this had long fallen to the wayside, perhaps in part due to better standards of living as a whole and more available food, through changing weather patterns, changing patterns of colonisation by the various Marshallese polities and clans, and better technology. Although the number of children born were still heavily limited- and necessarily limited, due to food and land shortage- this was probably through abortion or even abstinence (although, by all accounts, Marshallese society was not particularly unpromiscious, although neither did it fit the sometimes stereotype of the Pacific islands: it was not some free sexual paradise, there were societal, regulations surrounding sex and marriage, enforced by the entire community).

In times of desperation, young babies- no older- were likely killed, as a form of mercy killing and to reduce the burden on the community, although this was not systematic or ritualised, instead being similar to the high rate of infanticide and concealed births of the late Georgian era in England. Infanticide was therefore practiced for the sake of bwij, for the community, whenever the babies were too heavy a burden for the bwij, with all other burdens also ‘eliminated’: prisoners of war and the disabled (disabled children often being killed at birth) usually routinely and immediately (although eventually, when various iroij ro - sorry if I've messed up my Marshallese plurals- had the means to support the prisoners of war, prisoners could be kept as hostages, wives or labourers). Foreigners too were killed in times of desperation when otherwise they may have been supported.

Other practices besides infanticide and abortion existed to reduce the burden of caring for and raising children, the most important being adoption. A childless couple suffering from infertility would still be expected to have a child, but usually through adoption, with those having excess children or those who knew they were able to have a child and could therefore promise that child to an infertile couple, allowing their child or children to be adopted, with beautiful children naturally being highly prized, and adoption seen as an honour for not just the parents of the adopted child, but also for the biological parents who allow the child to be adopted in the first place. It enhanced the strength of the kinship system and strengthened family ties, as well as ensuring that no family was too heavily burdened by too many children.

To find a society where abortion was not practiced would probably be difficult. There are many natural abortifacients around the world, and of course physical methods of inducing abortion are known.