Does the US have legal obligation to intervene if genocide is occurring anywhere in the world?

NO.
There is no mandate demanding that the US MUST intervene in other countries.
In fact there are UN mandates that dictate that countries not intervene in the "internal affairs" of other countries, and that intervention without a UN resolution is a technical violation of international law.
However, if the US so chose, the president could order immediate intervention to stop the genocide, the Congress would need to be "notified" within thirty days (and approve or disapprove the action), the UN Security Council could be consulted, but a resolution wouldn't be required, as the US is one of 5 countries with carte blanche veto power, and any resolution condemning US action would most probably be vetoed.
NATO and SEATO and OAS treaties however operate differently, and within their spheres of influence, the US might or might not be required to act, if member states in those treaty organizations are threatened. Such was the case when the US invaded Grenada under the auspices of the OAS, the (modified) Monroe Doctrine and Truman Doctrine, to expel Cuban Army forces from the island.