In the lead up to the health care summit between the congressional Democratic and Republican leadership and the White House, the GOP plan and the President's plan were compared. It was an apples to oranges comparison, however, since the GOP plan was not fully fleshed out - not in terms of details but in scope. That plan (which has been taken down from the GOP web page) included an expansion of SCHIP, Medicaid reforms and malpractice reform, however it did not flesh out comprehensive reform which would have bent the cost curve.
The closest the Party has come to that was the John McCain proposal from the 2008 campaign, which would have provided a tax credit large enough for catastrophic health care, taxed all other health care coverage and provided for(presumably) tax free Health Savings Accounts. This reform would have effected every American rather than reforming around the edges. By providing a credit rather than a tax exclusion, however, it would have been more expensive if it were universally adopted - since no one would retain comprehensive insurance or employer provided insurance coverage - although most would not have been happy with such an option. Given the Republican allergy to progressive taxation, this plan would have either increased the deficit still further or would have required some form of payroll or consumption tax (maybe a Value Added Tax) to fund it.
This actually sounds like something I once proposed, since according to economic theory, such measures would be necessary to bend the cost curve (although the total price tag would be just as high as Obama care once you finished adding up tax credits and exclusions). How much it would bend the curve, however, is seriously disputable. Health care is not a "normal good." It is a good that makes consumption of other goods possible, at least in dire situations. You cannot buy a car if you are dead or disabled. Such non-normal goods do not respond well to supply and demand - especially if one has guaranteed access because one has an insurance card and a Health Savings Account/Flexible Spending Account. Except for elective procedures, there is absolutely no incentive to hold back on getting the best and most expensive care possible. I am also fairly sure that we do not want people to make such choices anyway.
Comprehensive insurance looks like it costs about the same as a catastrophic tax credit/HSA combination - at least from the fiscal standpoint - while not providing the same level of coverage.
Would such a plan be more popular than Obama care? Given that most people would be dropped from their comprehensive insurance as the tax exclusion for it goes away, I seriously doubt it.