Does the Constitution Prohibit Voting According Religious Beliefs?

The answer should be obvious, but all throughout the push to neuter marriage, people have said things that either explicitly or implicity indicate that those with religious beliefs about sex and marriage that are incompatible with those of the Left are not allowed to vote according to thier beliefs. The same goes for abortion.

To give you an example, on March 9, 2009 the Orange County Register printed a short letter from a Carey Strombotne that asked "If church and state are separate, why then do religious people have the right to weigh in on the issue of Proposition 8?"

The  paper subsequently printed several responses at this URL. (Note that the OCR now has a paywall in place.)

It is a perversion of the American system to suggest that citizens should not be able to vote according to their conscience. If that includes notions that are also found in tradtional, organized religion, so be it. Non-religious arguments have been made for the California Marriage Amendment and similar laws. Non-religious arguments have been made supporting the state's interest in licensing marriage as something uniting the sexes.

Ralph Alder of Santa Ana wrote:

A license from the state to marry is not a basic human right but rather a license granted on behalf of the people, because the people have chosen to issue it, just as with a driver's license, a business license or any other license.

And, as I've written many times before, individuals have access to the licenses regardless of their sex or sexual orientation, so there is equal access, even if one does not want to exercise that access.
Tom Culp of Dana Point wrote:

Does he mean that we have no free speech if our views are informed by religious conviction? Should we recuse ourselves from voting on such matters? How will this work in practice? Will there be a "Council on Separation of Church and State" set up by the government to determine what is political speech and what is religious speech? Will they then penalize newspapers for running op/ed pieces and letters to the editor that have a religious perspective?

Will there be a screener at polling places to question people on how they arrived at the votes?

  Good questions.  There were also letters from Mario Manriquez, Jr. of Cost Mesa, Michael R. Sumners of Santa Ana, Tom Farmer of La Mirada, Casey J. Durham of Mission Viejo, Ken Ellis of Anaheim, and Gene H. Cox of Mission Viejo.

Of course, someone could always say that we ought not vote according to our religious beliefs or our morality. But that "ought" implies it is immoral to vote that way - and such a statement commits suicide and should not be taken seriously, because it is a moral statement itself about how we should vote.
I vote according to what I think is right. Don't you? We may have a different basis for right or wrong. But unless you are willing to debate me about which one of us has a better basis to work from, then your plea for me not to vote according to what I think is right while you are voting according to what you think is right is laughable.
Read what the Constitution actually says on this matter:

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.

Voting for the bride+groom requirement in state marriage licensing does not establish an official national church. The people of, say, California are not Congress. If anything, the First Amendment and other Amendments protects a citizen's right to vote based on their religion. There is nothing in the Constitution that says an American can't vote according to their religious beliefs, whether they are voting at the ballot box, or as a member of a legislature, or as a Governor or President, or even as a Supreme Court Justice. As long as that vote does not adopt an official church or religious denomonation as the official one of the government, or deny someone their freedom of religion, or deny their freedom of speech or assembly, etc., it isn't violating the First Amendment.

The people who use the "You can't vote according to your religious belief" line are strangely silent when Leftist religious leaders and religious organizations speak up in favor of Leftism.