First, regarding federal government funding for any purpose, what happened to the sequester for which Obama cried that the sky was falling?
Next, with the exception of Obama spending money out of his own pocket, since Congress has the power of the purse, any tax-based funding that Obama provides to anybody must first be appropriated by Congress. But note that there are some major constitutional problems associated with Obama providing tax dollars for homosexual groups in other countries.
One problem is that given the Constitution's silence about gay and transgender issues, not only did the Founding States make the 10th Amendment to clarify that such issues are automatically unique state power issues, but Justice John Marshall had officially clarified that Congress is prohibited from laying taxes in the name of state power issues.
"Congress is not empowered to tax for those purposes which are within the exclusive province of the States." --Justice John Marshall, Gibbons v. Ogden, 1824.
But even if the states had delegated to Congress via the Constitution the specific power to tax and spend for homosexual issues, there's a big problem using tax dollars as foreign aid. More specifically, Justice Joseph Story had noted that Congress cannot use the General Welfare Clause (1.8.1) as an excuse to lay taxes for foreign aid.
"If the tax be not proposed for the common defence, or general welfare, but for other objects, wholly extraneous, (as for instance, for propagating Mahometanism among the Turks, or giving aids and subsidies to a foreign nation, to build palaces for its kings, or erect monuments to its heroes,) it would be wholly indefensible upon constitutional principles (emphases added)." --Justice Joseph Story, Commentaries on the Constitution 2 (1833).
So even if the states had granted Congress the constitutional authority to tax and spend for gay issues, the Oval Office can't give such revenues as foreign aid for any reason.
Sorry, sequester already mentioned. Long day at the office.