Same-sex marriage should be legalized because it is uncivilized to deny homosexuals their civil rights. The Constitution guarantees everyone equal rights and opportunities, however, s are denied both in terms of marriage.

is this better? I have support in other paragraphs

Yes, this is much better. :-)

Be sure to put a semicolon (;) after "opportunities" not a comma. Otherwise, it's a run-on.

Yes, this is a more detailed statement that provides reasons for legalizing same-sex marriage. You mention that denying homosexuals their civil rights is uncivilized and argue that the Constitution guarantees equal rights and opportunities for everyone, including the right to marry. If you have additional support in other paragraphs, it would be helpful to include them as they can strengthen your argument further.