NEW YORK (CNNMoney) -- Next move on Obamacare? It's up to President Trump.

Minutes after House Republicans shelved their bill that was supposed to save the nation's health care, Trump repeated that Obamacare was on the verge of collapse.

"Bad things are going to happen to Obamacare," said Trump, calling out the large premium increases and insurer defections that plagued the exchanges this year. "There's not much you can do about it. It's not sustainable."

While Trump is trying to shift the responsibility to the Democrats, it's his administration that will largely have to decide whether 20 million people who gained coverage under the sweeping 2010 health reform law will remain insured.

It's not at all clear that Obamacare is in a death spiral, but there's no question the program is troubled. Insurers have found themselves with sicker and costlier customers than they expected, forcing them to raise rates and exit certain markets.

But carriers say it's now largely in the hands of the Trump administration and Republican lawmakers as to whether they will participate next year. Over the past several months, insurers have urged officials to provide clarity on several key measures that they say will help shore up the exchanges.

"If Republicans want to stabilize the market, they have the tools to do so," Dr. Mario Molina, chief executive of Molina Healthcare, which has just under 1 million exchange enrollees in nine states, told CNNMoney. "If they don't act, they can't say Obamacare exploded. They made the decisions that led to people losing their coverage. They can't shift the blame anymore."

Insurers must decide in coming weeks whether they'll participate on the exchanges next year. At least one, Humana, has already said it won't. Molina said he will decide in May.

Among the top priorities is having Congress fund the cost-sharing subsidies that reduce the deductibles for millions of low-income enrolleee. Lawmakers have delayed their decision at least until April.

Also, insurers want the Department of Health & Human Services to clamp down on special enrollment periods so that people can't sign up when they become sick. And carriers want the agency to continue the Obama administration's efforts to bolster the risk programs that insulate them from costly policyholders.

"There's still a lot that can be done for market stability," said Kristine Grow, spokeswoman for America's Health Insurance Plans, a main trade association for insurers.

Trump officials have already started taking steps to stabilize the market, which they have had to do to fulfill their pledge not to have millions of people lose coverage as they tried to move to the Republican plan.

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