We have to make a distinction between the phrase "I have serious moral concerns about abortion" and "abortion is a subject which necessarily involves serious moral problems." If someone says the first, then I'll believe it is true for them.Now Cline makes it quite clear that when he says "problems," he just means "questions." And I don't know how anyone could deny that abortion involves serious questions. Surely the question of when something becomes a person with rights is a serious question? They involve questions, just look at what people say, so what reason can be given for thinking the questions unserious? While you might have an easy base case in, say, the morning after pill, the development to a full baby appears gradual, so even that base case becomes entangled in some difficult questions. Finally, some serious philosophers have argued that abortion is always or almost always wrong. You may think you have the right answer to these questions, or that some opposing views are irrational, but how are the questions not even serious?
The second, however, isn't a true statement. There can be cases where abortion poses serious moral questions, but not ever single instance of abortion does. The Christian Right benefits from a blurring of the distinction between the two because if they can get anyone to agree that any cases of abortion involve moral problems, they can quickly move to saying that abortion is inherently problematic. After getting agreement on the premise that abortion involves serious moral questions they then move to conclude that women can't make those moral decisions herself — and therefore they can't be permitted to legally chose to have an abortion.
If abortion opponents have offered the argument Cline ascribes to them--and he provides no evidence they have--the problem is in moving from "there's a question" to "we have the right answer."
This is part of a larger problem I've noticed--people think that philosophical questions are highly restricted, so the statement "it's a philosophical question" can be casually used as an important premise in an argument (the ghostwriting for the recent Antony Flew book comes to mind). Philosophy, far from being narrow, is about as broad in analysis as it gets. Philosophical questions are everywhere. What we need to stop the inference from "it's a serious philosophical question" to "I'm right."