The question isn’t whether he would try to overturn it, since the President doesn’t legislate, but whether or not he would sign or veto a bill to overturn it if it were to pass congress. Given his caving in on everything he formerly professed to believe in order to get the nomination, I kinda doubt he’s suddenly going to grow balls and say “you know what, I’ve decided we need to keep it”
There are certain things that the POTUS can presumably do himself with this law, though. I remember reading (don’t have a cite, so I could be wrong) that a President Romney would conceivably be able to issue an executive order permitting states to opt out of the ACA altogether if they want or an order prohibiting the IRS from enforcing the individual mandate penalty. I don’t know how feasible any of that is, however, given that (1) not every state is going to opt out (California won’t, Massachusetts won’t, Connecticut won’t, Oregon won’t, Hawaii won’t, etc.) and (2) an ACA absent the penalty would lead to complete chaos within the insurance industry, and I don’t know if Romney would pull the trigger on that.
The issue is whether Romney would issue those kinds of orders, or yes, whether he would push Congress to pass a repeal bill and sign it into law. I’m not so sure about either scenario actually materializing in a supposed Romney presidency.
This is the kind of thing that works well in debates.
“You used to support Civil Unions for gay Americans, now you’re against them. What will you position be next year?”
Oh gawd. I can’t wait to hear a moderator say something along the lines of that 
.
“My position is devolving!”
"I was retroactively against them before I was for them.’
Aw, shit. You know whatever question gets asked, Romney’s answer will be, “The American people don’t care about that, the only thing on their minds is the economy which has gone into the shitter under Obama. I’m not Obama, obviously.”
Are we not men? Are our husbands not men?