Any logical fallacy in Pete Buttigieg's rhetoric on abortion?

I think the fallacy you're thinking of is special pleading. If we trust the government legislature to delineate the moral/legal lines when it comes to self defense, why wouldn't we when it comes to abortion?

I have no idea what to say on this but I'm intrigued to read others comments. But I think thats the fallacy you're thinking of.

After going through the list of fallacies again, I think the Red Herring Fallacy might be a good fit of Buttigieg's rhetoric.

To quote this website, the Red Herring Fallacy is "Attempting to redirect the argument to another issue to which the person doing the redirecting can better respond" and "is a deliberate diversion of attention with the intention of trying to abandon the original argument." Its logic form is: (1) Argument A is presented by person 1. (2) Person 2 introduces argument B. (3) Argument A is abandoned.

Buttigieg's rhetoric fits this form: (1) Others are debating where to draw the line on abortion; each may have one's own specific argument (i.e. there are a category of arguments here), but all arguments in this category are about a generalized line in case laws or statutes. (2) Buttigieg introduces his argument that each pregnant woman should be entitled to draw her own line on abortion without resorting to any generalized line. (3) The implied message of Buttigieg's rhetoric is to abandon the whole category of generalized lines on abortion proposed by various people. However, since an individualized line may still work in conjunction with a generalized line (note that a line, or limit, in this context is a permission toward one side instead of a point on a scale, and two lines can be compatible if the individualized line is a subset of a generalized one), Buttigieg's rhetoric becomes deeply misleading and troublesome. I'm pro-choice, but I don't like such rhetoric, and it may backfire...

Does this analysis make sense?

While it is true the core here is merely an informal, unsupported claim—one can recognize an embedded, implied fallacy of false dichotomy or false binary fueling the position. 

This:  “That decision is not going to be made any better, medically or morally, because the government is dictating how that decision should be made.” 

The false dichotomy, false dilemma or false binary communicates a premise that erroneously limits what options are available. As such, the statement in question implies that an individual’s wishes and relevant legislation are locked in a zero-sum “mutual exclusives” dynamic. 

Furthermore, much evidence/data would be required to support the statement. However, political speeches are more about emotional language than facts and supported claims. 

There is no fallacy... it is just an opinion given.

The problem with the abortion issue is that there is no perfect or near-perfect analogy for abortion. It is a unique situation. Extremists on both sides refuse or fail to acknowledge that, and oversimplify the issue. I don't think Buttigieg is an extremist here but he is simply catering to his base with soundbytes.