A frequent occurrence these days, President Trump made his feelings known via Twitter after the Senate failed to pass the "skinny repeal" of the Affordable Care Act. And this time, he turned to what has become a familiar trope in his speeches about health care. Trump promised to let Obamacare "implode" in retaliation for Congress' failure to put a repeal and replace bill on his desk. He directly blamed the 48 Democrats and three Republicans who opposed the bill for its failure, ignoring the fact that the Republicans in charge of directing the process had proceeded in an overly secretive, completely unorthodox way.
Even before the skinny repeal was an option, Trump was already promising to "let Obamacare fail." He would have numerous ways to achieve that goal if he and his advisers decided it was politically expedient, which he seems to think that it is. “We’re not going to own it," he said last week. "I’m not going to own it. I can tell you the Republicans are not going to own it. We’ll let Obamacare fail, and then the Democrats are going to come to us." His reaction to the skinny appeal's failure presents essentially the same argument, by suggesting that Obamacare will fail on its own, without any mention of actions that he would or wouldn't take to push its failure along.
There are actual things that Trump can do to make the health care system under Obamacare worse. The most noticeable one, which has caused unending stress for health insurers and defenders of the law, has to do with Cost-Sharing Reductions, payments to insurers by the government to keep costs to consumers down. CSRs are not allocated directly by the Affordable Care Act, so technically, the executive branch does not have to pay them if it doesn't want to. So far, the Trump administration has been paying CSRs, but Trump is reportedly upset by this, and wants to end them. This would almost certainly lead to premium spikes.
Trump could also stop advertising Obamacare signups. The law requires plenty of people to sign up during open enrollment to keep the insurance pool stable, and normally this is accomplished along with a lot of ads during the open enrollment period. Immediately after taking office, the Trump administration cut ads for Obamacare sign-ups. And according to the Daily Beast, they (possibly improperly) funneled the money set aside for promoting sign-ups to market the law's repeal. If not enough healthy people sign up, premiums go up, because the insurance pool if predominantly sick, expensive to cover people.
Similarly, there is the potential for Trump to stop enforcing the individual mandate, the tax on people who do not have health insurance. On his first day in office, Trump signed an executive order that seemingly instructed the IRS not to charge people for not having health insurance. But it actually looks like the mandate is still being enforced, and it's not clear if Trump can actually stop it.
The changes Trump could make wouldn't get rid of Obamacare. It would still keep functioning essentially the same as it does now. However, they would lead to higher prices for those enrolled in it, and a less well-oiled health care system. If Trump can't muster the votes from his own party to change the system, it's tough to see how he benefits from it working badly.