During a speech on the housing market in Phoenix, Arizona, President Obama outlined a plan to repair the hurting industry. (Nicki Demarco/The Washington Post)

President Obama came here shortly after his first inauguration 41 / 2 years ago, pledging to arrest a tide of home foreclosures and to stabilize the deeply wounded housing market.

The next few years were a personal and political embarrassment for him: More people lost their homes, housing prices continued to sink, and economic growth was held back by the depressed real estate market.

But late in Obama’s first term, after he had upended his policies several times, the housing market started to rebound — and Obama returned here Tuesday to celebrate that recovery and insist that more work needs to be done.

“While it’s been a long, slow process that’s taken longer than any of us would like, we’ve helped millions of Americans,” Obama said at a high school here. “Now we have to build on this progress.”

The president outlined a series of policies he said would continue to boost the housing market, including a long-ignored legislative proposal that would allow more Americans to refinance at current low mortgage rates. He also said he is taking executive action to widen the pool of borrowers eligible to receive loans from federally backed programs; many borrowers without the highest-quality credit have been locked out.

But Obama also for the first time laid out specific principles for the future of the nation’s housing system, ideas that align him with bipartisan efforts in the Senate but sharply contrast with the views of many House Republicans.

The president said any future housing system must preserve the 30-year fixed-rate mortgage, a linchpin of the American economy that requires the government to provide guarantees to the housing market because bankers are unlikely to offer such long-term loans on their own. Thirty-year mortgages are rare in other countries, where government plays a far smaller role in housing.

But Obama also insisted that the federal government’s exposure to the housing market be more limited than it is today, when federal entities insure 80 percent of new mortgages. In particular, he called for an end to the federally owned mortgage giants Fannie Mae and Freddie Mac and said that any future system must place the vast majority of financial risk on private-sector lenders.

“No more leaving taxpayers on the hook for irresponsibility or bad decisions” by lenders, Obama said. “We encourage the pursuit of profit — but the era of expecting a bailout after your pursuit of profit puts the whole country at risk is over.”

Obama expressed support for bipartisan legislation being pursued by Sens. Tim Johnson (D-S.D.) and Bob Corker (R-Tenn.) that would eliminate Fannie and Freddie and replace them with a new federal agency that would provide insurance for new mortgage loans in exchange for a fee. Such an agency probably would preserve the 30-year mortgage.

House Republicans want to dramatically reduce the federal role in housing, and they complained Tuesday that the president has waited far too long to begin to discuss how to overhaul the system.

“For the past five years, the president has ignored the true causes of the financial crisis — the government’s misguided efforts at allocating credit through Fannie Mae and Freddie Mac,” said Rep. Scott Garrett (R-N.J.), a senior member of the House Financial Services Committee. “The president’s failure to address the largest bailout in our history has only emboldened Washington bureaucrats to continue the federal government’s domination of the housing finance market. Americans deserve better.”

The proposals unveiled Tuesday have long had support in the White House. But the administration avoided prolonged discussion of them because of opposition from House Republicans as well as a fragile housing market, where any discussion of change could derail progress.

The fact that the administration is now promoting an overhaul of the housing system reflects a revived confidence in the real estate market.

Obama pledged in February 2009 to spend tens of billions of dollars to rescue the housing market and save millions of homeowners, but his early efforts fell far short. Only a tiny fraction of the money he pledged was spent, and far fewer borrowers received help than he had pledged.

Obama routinely received letters from struggling borrowers saying that they couldn’t get help from his programs.

African American and Hispanic groups that were close allies of the president — and disproportionately affected by the housing crisis — were furious with him and his economic team for what they said were policies that were more sensitive to the needs of banks than borrowers.

But Obama and his team continually revised the housing relief programs. And while they have always fallen short of the goals he outlined here at the beginning of his term, analysts say they did make a difference, aided in large part by the Federal Reserve’s efforts to keep mortgage rates low.

The housing market has been rebounding since early 2012, with housing prices up 15 percent since then, according to the S&P/Case-Shiller index.