Sen. Ted Cruz, R-Texas, and Sen. Mike Lee, R-Utah, on Thursday introduced a bill that would allow states to define marriage however they see fit, even as gay rights proponents push legal challenges to traditional marriage laws in the wake of the Supreme Court's invalidation of the Defense of Marriage Act.

"Under President Obama, the federal government has tried to redefine marriage and to undermine the constitutional authority of each state to define marriage consistent with the values of its citizens,” Cruz said. “The Obama administration should not be trying to force gay marriage on all 50 states."

"The State Marriage Defense Act provides an important protection for states, respecting the right to choose for themselves how each will treat the institution of marriage under the law,” Lee added in his statement on the bill.

That's the same position Obama officially holds, per his May 9, 2012, interview with ABC in which he said he supported gay marriage "personally," but added that "he still supports the concept of states' deciding the issue on their own," as ABC summarized.

Similar legislation was introduced in the House in January by GOP Rep. Randy Weber of Texas.

Marriage has become a hot issue in Lee's home state, as a federal judge there overturned the state's gay marriage ban based on the Supreme Court's ruling in United States v. Windsor, which invalidated DOMA.

"The ruling makes Utah the 18th state where same-sex couples can legally wed," the Associated Press reported in December. The Supreme Court has issued a stay of the decision, though, and Lee and Cruz's press release only acknowledged that 17 states "have chosen to recognize same-sex unions."

The Justice Department opted to recognize the same-sex marriages certified by Utah in the time that elapsed between the the lower court ruling and the Supreme Court's stay.

“I am confirming today that, for purposes of federal law, these marriages will be recognized as lawful and considered eligible for all relevant federal benefits on the same terms as other same-sex marriages,” Attorney General Eric Holder said in a video released on the Justice Department's website. “These families should not be asked to endure uncertainty regarding their status as the litigation unfolds.”

Lee said, “How a state should define marriage should be left up to the citizens of each state,” in his statement on the new bill.

A Kentucky federal judge invalidated that state's gay marriage ban on Wednesday based on the DOMA ruling.

“Each of these small steps has led to this place and this time, where the right of same-sex spouses to the state-conferred benefits of marriage is virtually compelled,” U.S. District Court Judge John G. Heyburn II wrote.