This might be counter to the prevailing views here, but my response to the rhetorical question in the headline "Does religious liberty equal freedom to discriminate?" is a qualified 'yes.'

The American tradition of religious liberty includes a deep deference to the internal practices of religious groups. If a denomination like Roman Catholicism discriminates against women, and on the basis of marital status, when deciding whom to allow to become a priest, I think that is okay--and so does American law. A religious exemption for communion wine was allowed during the days of prohibition. If a denomination wants to bar openly gay clergy, or wants to bar same-sex marriage ceremonies from their churches, that is also within their rights under the "free exercise" of religion clause.

So yes, religious liberty does include a LIMITED freedom to discriminate.

But it is not a blanket right to discriminate. Other comments have covered the situations where religious liberty should and/or must defer to secular custom and law.