Weinberg and Sen. Raymond Lesniak (D., Union) last week introduced the bill, which Lesniak said would provide "an extra level of protection" after an October Superior Court ruling that allowed same-sex marriages.
While Gov. Christie's administration had appealed the decision, the governor dropped the appeal when the state Supreme Court said it would not halt marriages pending the case's outcome.
Christie previously vetoed a bill to legalize same-sex marriage. That bill included exemptions for clergy, as well as for religious organizations.
The court ruling legalizing gay marriage did not provide those exemptions, though Weinberg has said the First Amendment already protects clergy who object to performing same-sex marriages.
The bill introduced by Weinberg and Lesniak would allow only religious groups with event space solely for their members to turn away gay couples who want to wed there. Groups that let people not from their church use the space would not be able to deny same-sex couples.
Advocates with Lambda Legal and Garden State Equality, which was a party to the marriage equality lawsuit, have said they do not believe legislation is needed to ensure the already court-ordered right to marriage equality. Assembly lawmakers, meanwhile, had not scheduled any action on the issue, with a spokesman citing the strength of the existing right.
Weinberg said there are still unresolved issues surrounding same-sex marriage, including the status of civil unions and the state's recognition of out-of-state marriages.
Lawmakers will continue working with advocates, she said, adding that there was "no rush."