The law, commonly referred to as Obamacare, was plagued by issues in 2013. President Barack Obama promised that every American with insurance would be able to keep their plans, although many couldn't. The botched federal exchange website, healthcare.gov, was virtually impossible to use for weeks.

While the website is now functioning better, on the front end, support for the law and approval ratings for Obama are near all-time lows. Opponents shouldn't expect repeal anytime soon, Central Michigan University political science professor Orlando Perez said.

"As long as Obama is president, repeal is not going to happen, even if Republicans manage to win back the Senate (in the 2014 midterm elections), because there will simply not be enough votes to override a presidential veto," Perez said.

Chief among future concerns is getting younger, healthier Americans to sign up for health insurance through the law's insurance exchanges by the March 31 deadline. Doing so is key to keeping health care costs down as older, sicker Americans get insurance, potentially for the first time.

Younger Americans are considered less risky by insurance companies, so the Obama administration hopes that by adding them to insurance pools, the costs of adding older Americans will be offset. Doing so is key to achieving one of the administration's main goals with the law: leveling off steady, rising health care costs.

Republicans are looking to gain from the administration's health care issues in the November midterms, as they appear likely to make the elections in part a referendum on Obamacare. That might not work if young people end up signing up by the March deadline, CMU political science professor James Hill said.

"If the ACA overcomes its technical problems and the young sign up in sufficient numbers by spring, a positive message of the benefits of the ACA may blunt the only issue it appears the Republicans care about," Hill said. "That being said, it will be advantage Democrats. On the other hand, if problems or sign up issues persist, advantage Republicans."

Critics still hope for repeal

For the GOP, their attempts at getting Obamacare repealed might finally be seeing popularity. A December CNN/ORC poll showed the majority of Americans are now opposed to Obamacare. Just 35 percent of Americans who were polled said they supported the healthcare law, leaving 65 percent in opposition.

The key for the GOP might be to convince voters they have a plan regarding health care that provides an alternative or a fix to Obamacare, as Sen. Lindsay Graham, a top Senate Republican from South Carolina, told The New York Times.

"The hardest problem for us is what to do next,” Graham said. “Should we just get out of the way and point out horror stories? …You become a more effective critic when you say, ‘Here’s what I’m for,’ and we’re not there yet. So there’s our struggle."

Obama's initial healthcare plan called for a public option, a federal health insurance plan that would compete with private companies in an effort to expand access to health care and keep prices down. The White House and top Democrats were forced to drop that provision after fiscally conservative Democrats shot down the idea in the Senate.

Perez said a Medicare-for-all provision is simply not politically feasible, especially with a divided government.

"Politically, the Medicare-for-all idea does not have enough support to pass Congress," Perez said.