For six years now Republicans have promised to "repeal and replace" the Affordable Care Act. But while the House of Representatives has passed literally dozens of bills to repeal it (none of which actually got anywhere), they still haven't been able to come up with a replacement they can agree on. Last year House Speaker Paul Ryan said that the Republican Party owed the public an alternative proposal and promised that one would be forthcoming this year. To be fair, the year still has four months left to run, but there's no indication that any serious, detailed proposal is even in the works.
Actually, there is a Republican proposal they could offer. It was outlined by the conservative Heritage Foundation, and a version of it was pretty successfully passed and implemented by Mitt Romney when he was governor of Massachusetts. Under this plan the existing Medicare, Medicaid, military, and veterans' healthcare systems remain intact, but otherwise the system emphasizes free-market solutions and individual responsibility and choice.
Of course, the Republicans can't propose this as a replacement for the Affordable Care Act because it actually is the Affordable Care Act. The ACA does have some additional features -- among other things making a lot of preventive services under Medicare free, greatly improving the financial stability of the Medicare Trust Fund, and remarkably at the same time reducing the federal deficit -- but the basics are the same. Romney's far-right opponents made a lot of noise about that in 2012, calling the Affordable Care Act "Obomneycare."
The GOP might have been better off politically had it simply congratulated Obama for adopting a Republican proposal. But as I've documented previously, a bunch of Republican opinion leaders are on record as advising total opposition to any healthcare reform backed by Democrats, no matter what it might be, for fear that if it made things better, in the long run it would give Democrats an electoral advantage. That's not some cynical inference I'm drawing; it's what the leaders in question have said explicitly (link) and often in print. I've checked the original sources to verify that they're legitimate quotations and not Internet rumors, and I urge you to verify that for yourself.
Incidentally, none of this is to say the Affordable Care Act couldn't be improved. As I said in a post four days ago, it has its problems that ought to be addressed. In fact most of them probably would already have been dealt with if the current Congress weren't so dedicated to blocking anything Obama might get credit for.