Can the Republican Party write another bill later to try to repeal the Affordable Care Act again? Or is it here to stay?
Theoretically, the Republicans can try to repeal it again, as many times and for as long as they want - or at least until they get thrown out of office by mad voters.
In practice, the longer it's around, the more political capital will be required to repeal it, and thus the less likely it'll get repealed. Republicans standing against their own camp on grounds that their own voters might lose their insurance is a telling sign that this is occurring today.
That being said, what the administration could also do - and in fact, already started to do - is try to sabotage it in some way or another. For instance by trying to defund it somehow, by adding a few layers of red tape, etc. The point being, if they somehow manage to set things up so the ACA is almost guaranteed to look like a failure (if only by a few criteria) a few years down the road, then they could gather enough momentum to try to repeal it again when it does. (Or maybe, as Trump tweeted, it'll implode all by itself.)
No, the Republicans are able to introduce further bills or amendments to repeal or replace the Affordable Care Act.
This week's vote is in fact the second attempt by Republicans at healthcare reform, after the first attempt failed in March this year due to opposition from the Freedom Caucus.
However, President Trump is able, and has already tried, to weaken some provisions of Obamacare through executive action. This article by The New York Times shows some of the actions that Trump has taken.
Additionally, Executive Order 13765, titled "Minimizing the Economic Burden of the Patient Protection and Affordable Care Act Pending Repeal", was signed by Trump on his first day in office that aimed to weaken regulations and procedures associated with Obamacare and called for its prompt repeal. The full text can be found here.
It directs the secretary of health and human services, as well as other agencies, to interpret regulations as loosely as allowed to minimize the financial burden on individuals, insurers, health care providers and others.
No, they will be able to attempt to repeal the ACA again in the future, but it does become significantly more difficult.
What was significant about this recent vote is that they attempted to repeal it through a process known as budget reconciliation. Reconciliation only requires a simple majority (50 votes) to pass and disallows filibustering, but they can only consider a reconciliation bill once per year.
To attempt to repeal the ACA now without waiting another year, the senate would need to go through the regular process to pass a bill. But that requires a supermajority (60 votes) which Republicans don't have, and any bill can be filibustered by democrats.
ACA is not 'safe' under any circumstances. It was already in trouble in its current form. Aetna bailed because it was losing money on the exchanges.
The NYT also chimes in with serious problems, and this was mid 2016 while the presidential campaign was well under way, with Clinton the presumptive front runner.
So the fact that the repubs couldn't agree on a replacement, and the fact that they are making the same mistake the dems made - putting together major legislation while ignoring half the country - doesn't make the current ACA 'safe'. They can try again, if they choose.
This problem won't be fixed as long as the discussion revolves around excusing one's chosen political party, while looking to slag the other. It will be solved when everyone involved makes solving the problem the main priority, regardless of who gets flamed or who gets the credit.
The chances of that happening are not good. The majority of politicians, and a good deal of citizens, are too focused on a partisan battle that no one can win.
Firstly, it appears to me that ACA is failing on its own. Actually, when it was passed it was tremendously flawed and because of the lack of any bipartisan support, wasn't corrected before (as well as after) it went into effect. It has been withering on the vine since Obama signed it into law. The enormous price hikes and Insurer's withdrawals from the market are both signs of serious (and I expect, fatal) problems. All Trump needs to do is to direct the IRS and the DOJ not to enforce the requirement for all to be insured. I'm not clear how easily this can be done, the IRS doesn't appear to have done much to sabotage it, so far... To answer your question: I think that the Legislative Branch is going to have to eventually either drastically change it or replace it in the not too distant future. I'd guess by 2023 or 4.