The Constitution lacks any direction quotation of the phrase, however the first part of the First Amendment says "Congress shall make no law respecting an establishment of religion". This means that the United States cannot make any laws based on religious beliefs or claim that the US is a nation based on religion.
However, the saying is a reference to the first lines of the First Amendment "Congress shall make no law respecting an establishment of religion". This was created to "separate Church and State". Even though it never says the exact words, that is what part of the First Amendment basically does.
The answer is no. The phrase "Separation of Church and state" has never been in the Constitution of the US. In fact, just the opposite. The Law back in the day stated that if it were beneficial morally to the community, the church had influence over state. Also, people who site the phrase don't realize that this was in a personal letter by Thomas Jefferson and that it was never a law.
"Congress shall make no law respecting an establishment of religion" is taken Establishment Clause in the first amendment which is in the constitution. There has never been any separation of church state that is pleasurable due to the fact the that since the beginning, people have been pushing their religion and imposing it onto the government to make laws against certain things. It is unconstitutional for the government to make a law against something in which respects a religion or any religion. I.E. Saying "God made Adam and Eve, not Adam and Steve" is perfect example of how there is no separation. Government is not allowed to make a law due to a religion therefore, taking a religious moral from the bible, Qu'ran or any holy book, is unconstitutional. Decades ago, when interracial couples wanted to get married, they had people saying it was immoral due to their religion and it's an abomination. This is, yet another example of how there is an extreme lack of Separation of Church & State. So any government official that says they won't legalize something because it's religion, you can have fun in calling them out for being unconstitutional.
Christians claim that there is religious oppression by us saying that they cannot use their religion to control the law; because not everyone in this country agrees on religion. However, there are still only 11 states that say "gay is okay", because much of our country doesn't want to look a the facts. Only at the Bible, which also says that if a woman who is not a virgin gets married, she should be killed, and that if you get divorced, the wife is to be stoned. How many Christians follow that? I'm assuming very, very few.
Bottom line : our country was founded on religious freedom, but there seems to only be freedom for Christians. Ultimately, laws should be made for their improvement on the general public's lifestyle, not because your religion says so.