Yes, and a qualified no. International law and domestic policy are separate. And complicated.
According to the UN Treaty Database, Obama signed the Paris Agreement on April 22, 2016 and accepted it on September 9, 2016. These words have specific legal meaning.
Signatures are subject to ratification, acceptance, or approval. Obama's signature signaled the US intention to comply with Paris Agreement once in effect, but not obliging the US to do so until formally ratified, accepted, or approved by its own domestic process.
Obama then sends an acceptance letter. What is this? And how is it different than ratification or approval? Let's define the terms first:
... a state indicates its consent to be bound to a treaty if the parties intended to show their consent by such an act.... The institution of ratification grants states the necessary time-frame to seek the required approval for the treaty on the domestic level and to enact the necessary legislation to give domestic effect to that treaty.
The instruments of "acceptance" or "approval" of a treaty have the same legal effect as ratification and consequently express the consent of a state to be bound by a treaty. In the practice of certain states acceptance and approval have been used instead of ratification when, at a national level, constitutional law does not require the treaty to be ratified by the head of state.
But wait, Article II section 2 says the President can sign treaties only with the "advice and consent" of the Congress:
He shall have power, by and with the advice and consent of the Senate, to make treaties, provided two thirds of the Senators present concur
This is a treaty, and the US Constitution says the President requires advise and consent, so how could Obama just sign it?
Turn now to the Center for Biological Diversity Climate Law Institute, which drafted a 2009 working paper titled "Yes, He Can: President Obama’s Power to Make an International Climate Commitment Without Waiting for Congress". In this working paper, CBD identified a two prong legal footing on which Obama could sign, apparently without domestic ratification, the Paris Agreement:
- The Clean Air Act, as amended, of 1990, which provides that
the President may enter international agreements... and develop standards and regulations which protect the stratosphere consistent with regulations applicable within the United States."
- The Global Climate Protection Act of 1987, which provides that the President may "identify technologies and activities to limit mankind's adverse effect on the global climate" by "slowing the rate of increase of concentrations of greenhouse gases in the atmosphere in the near term" and "developing and proposing to Congress a coordinated national policy on global climate change." This legislation also states that the "Secretary of State shall be responsible to coordinate those aspects of United States policy requiring action through the channels of multilateral diplomacy, including the United Nations Environment Program and other international organizations."
So, Obama accepted the Paris Agreement on the good faith basis that he had already had the domestic authority to commit the US to international accords. That acceptance is legally binding under international law.
Beside upholding the obligation, the US could ignore, denounce, or withdraw from it. The last two have specific, legal meaning, but all three boil down to the same thing: nothing happens. Unfortunately, the Paris Agreement has no teeth, no provisions for punishing those who agree to uphold but ultimately don't uphold. In this regard, Paris is more like a recommendation than a contract.
So, yes Trump could have the Secretary of State draft a notice of denunciation, inter alia, and send it to the UN, effectively signaling the end of US international cooperation. But, no, short of legislation change, he can't really stop the US paricipating in the essence of the Paris Agreement: the Clean Air Act and Global Climate Protection Act proxy the agreement's fundamental environmental and economic aspects.
To answer your last question, the Paris Agreement went into effect in November. Even if the US defaulted, the treaty would continue to be in force, per Vienna Convention on the Law of Treaties, as amended