That is because the previous president decided to unilaterally bind the country to the agreement rather than seek treaty ratification from Congress. It is a risk that ruling by executive decree entails.
As a country we need to be able to have responsible, sober debates and actually vote through policies instead of just working through whoever happens to be in the Oval Office. It's an unstable way to do things.
That said I would argue it's mostly unstable because the Republican party has been hijacked by its extreme right wing. It's astonishing to see that even with control of both houses they can't internally agree on policy enough to get things done. Much less find some sort of bipartisan compromise/consensus. Dems have their problems too, but I think the bulk of the responsibility lies on the very hard right impulses of the likes of the Freedom Caucus and the Ted Cruzes of this world.
No, pretty much zero Democrats see increasing the scope of government as principle. You might plausibly argue that most Democrats agree with the statement that there is at least one area where government needs to expand it's size, role, or influence (though they often wouldn't agree on where or why.)
Then again, most Republicans, even much of the Freedom Caucus, agree with that too, which is why despite small government rhetoric they often vote for expansions in defense, law enforcement, and security spending, staffing, and/or powers.
The USA cannot be trusted as an ally. That is the takeaway here. Saying that "but Obama's worse than Trump!!!" isn't changing that and is entirely besides the point.
ETA: And, for what its worth, the Senate ratified NATO, and that hasn't stopped Trump and Republicans from doing their utmost to sabotage it. It's laughable for folks to pretend like this is some deep matter of principle.
Congressional Republicans don't like it, but given the set of options they have they choose to allow Trump to take the hatchet to NATO. Standing in the way of that would prevent them from focusing on priorities they place higher than the Transatlantic Alliance.
Which is a key reason why international cooperation is so dangerous. It's insensitive to domestic politics and law.
Kind of a moot point IMO, given the current state of the GOP. Had Obama gone to Congress they probably would've reflexively voted it down anyway. The current GOP is willing to damage America's trustworthiness on the world stage and this is more proof to the point.
I don't know if most mainstream Republicans would've done something as drastic as this after the accord had been agreed to but Trump is basically the head of the party now so here we are.
The difference in the climate today is that one party was essentially saying no to anything Obama might do. Even if they privately agreed or could find some ways to agree with him, they knew that if they so much as smiled in his direction they'd get primaried and kicked out. That climate poisons the relationship between the branches, and gets you to non-standard processes which we can all agree are sub-optimal.
For a balance of powers system to work you need to actually have a working relationship between the branches. It's not going to work if one says, "NO!" then expects the other branch to take their toys and go home. In the real world, we all have to accept compromise toward our objectives, and if we don't we'll get disaster for all our objectives down the line.
The problem is the gross disconnect between Local, State, and Federal interests. Polls have support for the Paris agreement well into the majority, beyond the margin for error; were it to head to something like a referendum, the US would have signed onto the Paris agreement.
And even that being the case, President Obama was well aware of the fact that it never would have been ratified by the Senate.
The international credibility problem also reaches critical levels when you consider that the United States has 4 year terms but signs onto 20 year agreements, when the political sphere is so radically split. Every four years, the International community is forced to hold its breath as one of the top world superpowers seemingly flips a coin as to whether or not it will uphold its bargains. It is worth noting that this wasn't always the case; it was standing tradition that you followed through on a previous administration's agreements, simply to maintain credibility.
Whether it is a function of Trump himself, his die-hard base, or a large group of confluent factors (my vote is for this one), this tradition of maintaining credibility despite political cost holds very little water with the current administration. Worse, this is exactly what his base seems to want and clamor for. They are painting a political climate where the country takes a 180 degree turn (rather than, say, a 120 degree curve) every 4 to 8 years.
How long can a country in a global economy at the scale of the United States maintain that strain without snapping? It'll be interesting to watch, if nothing else.
Other countries such as Brazil had the treaty ratified by their senate.
Arguing that we now have a treaty that allows us to make other treaties without following the Constitutional process ... well.