The battle royale for the hearts and minds of voters on President Barack Obama's signature health care reform law kicked into high gear mere moments after the U.S. Supreme Court announced that it had been upheld.

Republicans instantly pounced on the ruling as proof that Obama is trying to raise taxes, while Democrats seemed eager to put the battle behind them and move on.

Millions of dollars have been spent to shape public opinion on the law, and the campaigns and deep-pocketed donors to super PACs will spend millions more. But political experts say the cash poured into messaging may do little to sway voters, many of whom already hold deeply entrenched views on the law.

Instead, the fight for undecided voters may depend heavily on the Democrats' ability to distill a somewhat complicated message into digestible sound bites and Republicans' ability to force-feed a message that the law amounts to a tax increase.

"What we're waiting for in the aftermath of the decision is to see whether the decision legitimizes or makes the health care policy more popular," said Nathan Gonzales, deputy editor of the Rothenberg Political Report. "Right now, it's very polarizing. If health care is the top issue for a voter, they've probably already made their mind up, and the ongoing message war won't change how they feel about it. We have to see for moderate voters if the Supreme Court decision persuades how they feel about it. How the legislation is framed will go a long way into how they feel about it."

Democrats, especially Obama, who was handed a legacy-defining victory by the court, were obviously buoyed by their legal win.

"Essentially, Obama will use the health care decision to brag a bit to the base: 'You see, hope and change mattered. The court decided for us despite the doom and gloom.' It'll make them feel good. I don't think he'll dwell on it because it's unpopular," said University of Virginia political science professor Larry Sabato.

Courtroom triumphs are a far cry from winning in the court of public opinion, and in that regard, Obama has a lot of work to do.

According to a CNN/ORC International poll conducted last month, 51% of Americans oppose the law -- most because they think it goes too far but some because it doesn't go far enough -- while 43% are in favor.

Those findings dovetail with a Pew Research Center poll that asked respondents whether they would be happy or unhappy depending on how the court might rule. It found that 44% of voters said they'd be happy if the entire law were thrown out, and 40% wanted to solely chuck the so-called individual mandate. Just 39% said they'd be happy if the entire law was upheld.