Probably the most well-known secular argument against abortion is by Don Marquis. The paper is called Why Abortion is Immoral (sorry I don't have a link to a non-paywalled version) and the argument goes roughly like this:
Start by asking why death is a misfortune. Marquis' answer is that it cuts us off from all of the potential value in our futures. This is why it's worse than being robbed or injured. At least if I'm robbed I still have the hope of a worthwhile future, even if that future has been diminished in some ways. But a fetus is a being like us in this very respect: other things equal, it has a future with a potential for value of the very same kind that makes death a misfortune for beings like you and me. This is true even if we believe that the fetus isn't yet a person. And so ending the fetus's life does it a very great wrong: it robs the fetus of the possibility of a valuable future in just the way that killing your or me does. In short, abortion is wrong for exactly the same reason that killing a full-fledged person is.
That's the argument in brief. Notice that it doesn't claim that all abortions are wrong. For example: if a fetus has defects so severe that it wouldn't be able to have any conscious experience, Marquis' argument doesn't say that abortion is wrong. Likewise, if the fetus's life would be a life of unrelenting pain, Marquis' argument doesn't rule out abortion. But if it's a matter of a baby being born into poverty, that's not enough to justify abortion on Marquis' approach. It's clear that many, many people who are born poor and live poor still have lives worth living.
As with any philosophical argument, there's room for objections. For example: Marquis' way of arguing suggests that how wrong it is to kill someone depends partly on their age, with killing the elderly being less wrong: the older person has less potential for value in her future. But that's not a knock-down objection. For one thing, Marquis doesn't have to claim that the sheer quantity of potential value in one's life is the only thing that bears on the wrongness of killing. All he needs to say is that it's a very weighty consideration.
Whether it's an objection or not, another interesting question to ask is whether Marquis' argument shows more than it might have been intended to. In particular, If the argument is a good one, then essentially the same argument would show that most of our meat-eating habits are morally unacceptable. If so, people who object to abortion but not to eating meat have a serious consistency problem.
I think Marquis' argument does offer plausible non-religious objections to abortion. Whether the argument is good enough to justify stringent legal restrictions on abortion is a further question, but even if the answer turns out to be no, Marquis has shown that the "no" doesn't come quite as easily as the pro-choice side sometimes assumes.