Recognition of foreign governments is normally shown by the establishment of a consulate, embassy, or special mission to the foreign government in question. These embassies (etc) are established and run by the US Department of State, headed by the Secretary of State (currently Hillary Clinton) and overseen by the President. Therefore, recognition of foreign governments is typically done by the executive branch.
Congress has the power to ratify treaties with foreign governments and can pass legislation involving the recognition of or relationships with foreign governments. So sometimes, recognition of foreign governments can be done by the legislative branch.
But generally speaking, recognizing foreign governments is a power of the Head of State of a nation, and the Head of State of the United States is the President, so it's an executive branch power.
Executive Branch :)
legislative
Congress
The Executive Branch (President) hears cases between the state governments.
Recognition of new foreign countries and governments falls under the purview of the Executive Branch. The Department of State advises the president, and it is the president who has overall responsibility for forming and executing the foreign policy of the US.
The legislative branch
The Judicial Branch decides whether or not a law is unconstitutional.
The legislative branch (Congress) decides on what should be law and makes the laws.
The judicial branch.
The Judical Branch
The Executive branch decides is a treaty is unconstitutional
The judicial branch decides if laws are legal and it's called Checks and Balances.