In another development, House Republican leaders announced Thursday that they plan a floor vote to repeal the bill during the week of July 9. The vote is basically a political statement; while the repeal vote should easily succeed in the Republican-led House, there's virtually no chance it'll get past the Democratic Senate.
But President Obama said the country can not "go back to the way things were" or "fight the political battles of two years ago."
"Today I'm as confident as ever, that when we look back 5 years from now, or 10 years from now or 20 years from now, we'll be better off because we had the courage to pass this law and keep moving forward," he said.
He added he didn't pass health care because it was politically popular, but said that the mandate was "supported by members of both" political] parties, "including the current Republican nominee for president," referring to Romney's passage of an individual health care mandate while he was governor of Massachusetts.
"I did it because I believed it was good for the country. I did it because I believed it was good for the American people," the president said.
As for other parts of the law, he said insurance companies can't drop people if they get sick, are required to provide free preventative care, and allow young adults up to age of 26 to stay on their parents' health insurance plan.
For his part, Romney said he disagrees with the Court's decision to uphold the health care mandate, requiring every American to purchase health insurance, by vowing to "repeal and replace" it as president.
Standing with the U.S. Capitol as a backdrop and a sign on the lectern that said "Repeal & Replace Obamacare," the Republican presidential candidate said, "If we want to get rid of Obamacare we have to replace President Obama."
"What the court did not do on its last day of session I will do on my first day" as president, Romney said. "I will act to repeal Obamacare."
, with Chief Justice John Roberts siding with four liberal justices, that the federal government does have the power to require all Americans to acquire insurance if the "fine" imposed on those who don't is considered a tax.
Romney and Republicans are using the court's specification that the mandate is a tax levied on Americans as Democrats' recklessness to impose more taxes during a difficult economy.
"Obamacare raises taxes on the American people by approx $500 billion," Romney said.
Romney also reused many of the arguments popular during the height of the health care deliberations: that it kills jobs, cuts Medicaid and puts the government in between doctor and patient.
Romney said this election about a "choice" between "a larger and larger government... or whether, instead, you want to return to a time where the American people will have their own choice in health care."
Watch CBS News' special report.