That's hard to say. With the current makeup of the supreme court, it's likely they'd simply declare any law protecting abortion rights as unconstitutional because mumble mumble and get away with it. But what's preventing them from doing even that is that Republicans (thanks in large part to politicised redrawing of district boundaries) have a majority in one of the two legislative bodies, so the Democrats couldn't pass that protection regardless.

So likely the minimum that's needed to codify abortion rights would be a Democratic majority in both legislative houses and a Democratic president.

On the topic of coalitions: The US doesn't have coalitions in the ways many other countries have, partially because of the way the president is elected. Voters have a separate item on their ballot to elect (electors who will then vote in the electoral college for) the president. The way this occurs is through first past the post, where the largest portion of the votes (even if a minority) gets all the electors in that state (except in Nebraska and New Hampshire, where the state breaks it into districts). I'm in Michigan, for example. In 2016, Donald Trump got 47.5% of the vote in Michigan to Hillary Clinton's 47.3% and thus got all 16 of Michigan's electoral votes (out of 538). Had 11,000 more people voted for Clinton (let's say, by not voting for the Green party), she would have won Michigan's electoral votes, which is a 3% swing in the electoral college, but given that most states are pretty much guaranteed to go one way or the other (e.g. Indiana is a safe Republican state while neighbouring Illinois is a safe Democratic state), those 11,000 votes would be massively influential. This is why "swing states" are so stupidly pivotal in US elections.

So because of all of that, there's not an option for the Greens to join a coalition, even if they wanted to (which I don't think they would, as the US Green party is currently under the control of a Russian asset and it's well known that Putin wants a Trump victory).

The American electoral system is ridiculously, stupidly backwards and basically designed to empower certain people over others. If there were a parliamentary democracy here the US, and probably the world (given the US's love for foreign intervention), would be much better off.

Could the Green party and Democrats form a coalition and choose the President accordingly

They certainly could, but why would they? Not only democratic party stand for a lot of things greens find unacceptable (and vice versa), but disproportion between both parties is so huge that greens would at best got given some paltry compensation (and a huge bill of firming democratic party atrocities with their names, this would essentially be their end) and most likely just become completely ignored and cut off after election.