WASHINGTON (Reuters) - The U.S. government is working to limit the diplomatic fallout after Texas executed a Mexican national over Washington's objections that it violated U.S. treaty obligations and put U.S. citizens abroad at risk.

Secretary of State Hillary Clinton was "quite disappointed" on Thursday after Texas went ahead with the execution by injection of Humberto Leal Garcia, convicted of murdering a 16-year-old girl, despite pleas from the federal government for a last-minute stay, State Department spokeswoman Victoria Nuland said.

"It's important that our partners overseas know that the U.S. government, the executive branch, was not comfortable with what happened in this case," Nuland said.

"The secretary is making clear to her counterparts, whether they're in Mexico or anywhere else, that we seek to remedy this situation and we seek to remedy it as quickly as we possibly can," Nuland said at a news briefing.

Nuland said the Obama administration would work with Congress to speed passage of legislation that would spell out the rights of foreigners to consular access.

The top U.N. official for human rights, Navi Pillay, issued a statement on Friday saying Leal Garcia's execution "places the U.S. in breach of international law."

The U.S. Supreme Court rejected a stay of execution despite warnings from the Obama administration that the case violated the Vienna Convention on Consular Relations because Leal Garcia had not been given appropriate consular access.

In legal briefs, the U.S. government had warned the execution would create an irreparable breach of international law and Mexico's government said it would seriously jeopardize cross-border cooperation on joint ventures and extraditions.

Leal Garcia, 38, who had lived in the United States since he was an infant, was convicted of raping the girl and bludgeoning her to death with a piece of asphalt in 1994.

While the United States has joined the Vienna Convention, critics have argued that individual U.S. states are not bound by it until Congress passes enabling legislation.

Nuland said the State Department was concerned that the case might impact the welfare of U.S. citizens who run into legal problems overseas.

The State Department says some 5 million U.S. citizens live overseas and many millions more travel regularly outside of the country. About 3,500 Americans were arrested overseas in 2010 and U.S. consular officials conducted more than 9,500 prison visits.

"Frankly, if we don't protect the rights of non-Americans in the United States we seriously risk reciprocal lack of access to our own citizens overseas," Nuland said.

"Texas justice is Texas justice. This is simply about ensuring a non-American facing judicial proceedings in the United States has the same rights that we expect an American facing judicial proceedings overseas would have."

(Editing by Bill Trott)