Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances.
The first two phrases are intended to protect religions freedoms of U.S. citizens and other persons entitled to the protection of US law. One fear, as most history students know, was the establishment of a State religion. Supreme Court decisions, not involving foreign assistance, have said this language means the U.S. Government cannot get "excessively entangled" in religion. This means, for example, that prayer in public schools, or posting of the Ten Commandments in public buildings, is prohibited, etc.
There is a strong body of literature explaining the history of the First Amendment cases and their application, and it is not the purpose of this article to fully rehash or summarize this literature. If that were my purpose, I believe there is almost a self-evident case that whatever the Executive Branch might do in it's foreign aid program, it would not be "Congress [making a law] respecting an establishment of religion." I will limit the legal discussion, therefore, to the following four paragraphs.
Author Barry Adamson, in his book, Freedom of Religion and the First Amendment and the Supreme Court (Pelican Publishing Co 2008), has given a masterful and very detailed analysis of the meaning of the term "establishment", and the history of this particular word in the First Amendment. Unlike today, the term was used in the late 18th century predominantly to mean an "institution" of religion. Most of the colonies had already enacted their own versions of the establishment clause, in a reaction to public outrage over the state government support for various sects or factions especially the Congregationalists (in the north) or the Anglican churches (in the south). Many of these same colonies, in the campaign to ratify the Constitution, were promised by one James Madison and others, that an equivalent provision would be considered as an amendment to the U.S. Constitution incorporating essential rights. We know now the outcome as the first 10 amendments. There is no question but that it was aimed only at the preventing the establishment of a national, government sanctioned religion with compelled attendance, tithing, privileges concerning public office, etc. It achieves this by saying, elegantly and simply, that Congress would make no law about any such institution, coupled with the free exercise clause which immediately follows. It said and meant nothing about the Executive Branch. Moreover, virtually the same group of individuals, at almost the same time, approved a number of laws indicating they did not understand the term to mean government and religion should be hermetically separated, but rather that no particular sect should be favored by government. For example, in August of 1789 they delivered a bill to President Washington (the Northwest Ordinance) which declared that "religion, morality and knowledge, being necessary to good government . . . shall forever be encouraged." It was made a condition to the admission of future states, that each have a constitution consistent with the principals therein expressed. They passed the 1789 Judiciary Act requiring all federal judges to take an oath followed by "So help me God." And they approved in September of 1789 a Thanksgiving Resolution in which they requested a national day of thanksgiving to "Almighty God." In later years, Congress would approve coinage (1864), a national anthem (1931) and motto (1956) with "In God We Trust". A national anthem was approved in 1931 with "In God is our trust", the pledge of allegiance phrase "one nation under God" was signed into law in 1954. It is a safe bet that virtually all of the Founders, and certainly the author of the clause (James Madison), would have been, say, bewildered by a 9th Circuit decision in 2002 that the words "under God" in the pledge of allegiance violate the establishment clause. That court was overruled, although other cases now may be pending on this same point.
The modern extension of the establishment clause to actions of the Executive Branch first happened in 1947, in the case of Everson vs. Board of Education . It had earlier been made applicable to the States also, via a between-the-lines reading of the 14th Amendment as a type of protected "liberty" interest.  It has evolved now into a doctrine to prevent, among other things, "excessive entanglement" of the government with religion sufficiently ambiguous as to give courts great leeway to come up with a variety of decisions, turning (primarily I submit) on their own varying degrees of respect for historical traditions. Lemon v. Kurtzman . Most of the reasoning derives from a famous phrase in a letter one Thomas Jefferson wrote in 1801 to a Baptist group in Connecticut that wished to gain his favor in their own efforts to support a state "disestablishment" clause in that state. Everson and latter decisions take as evidence of the Founders' intent a reference to the building of a "wall of separation between Church & State." This was his phrase, not Madison's nor the Constitution's. Jefferson lived in France from 1784 to 1789 and was not involved in the drafting of or the debate about establishment clause, though he did have a hand in drafting early versions of a free exercise clause for the Virginia colony's constitution.