Interesting, thank you for that information!
I guess this is quite a subjective exception to the Tenth, so theoretically a judge could swing either way on whether abortion falls under its remit. They already tried to bring in a firearms law under this clause and it was thrown out at supreme court.
My interpretation of both the law and what people have said here (thanks also to Labraradabrador) is that any ruling on abortion would be very difficult to write into federal law, partly because it would have to be extremely carefully crafted to avoid the fate of Roe v Wade but also because of unpredictable resistance / support from individual Democrats and Republicans, and because actually passing the bill would involve getting it through the House which is Republican controlled. Furthermore it relies on an exemption that may be challenged by future judges.
So theoretically yes it's possible, probably a mistake to base a re-election promise on (tbh, I don't see the Democrats getting control of House, Senate AND Presidency -- I think the country is way too polarised right now), and I think Biden is well aware he is a) not very popular even with his own party and b) this happened under his watch, even if it wasn't his fault.