that depends on your definition of "interest". Should government be obligated (as in they're liable) to protect all innocent life, probably not.
Should government be organized so as to protect innocent life where possible, probably. Although, the degree to which individuals trade liberty for security is debatable.
If you're suggesting that government should intervene with the medical decisions of individuals (specifically whether they should or should not terminate a pregnancy) in order to protect the innocent life of the unborn child, then I'd argue that there are other more positive ways to accomplish that goal, short of coersion.
What is the purpose of government?
Let's start there.