The ban, enacted by state voters in 2008 as Proposition 8, was overturned in February by the 9th U.S. Circuit Court of Appeals in San Francisco.
The measure's sponsors last month appealed to the U.S. Supreme Court to grant review of that decision and uphold Proposition 8.
But in briefs filed on Friday, two same-sex couples and the city of San Francisco argued there is no need for the high court to take up the case because the 9th Circuit ruling was based on narrow grounds and was "a careful and straightforward application" of Supreme Court precedents. "This court's traditional standards for the exercise of certiorari jurisdiction lead inexorably to the conclusion that this court's review is not warranted," lawyers for the two couples wrote.
A high court decision to hear an appeal is known as granting certiorari.
The court is expected to announce in late September or early October whether it will consider the Proposition 8 case.
The Supreme Court grants review of less than 5 percent of the cases appealed to it. It generally selects issues that are either of great constitutional significance or have been the subject of conflicting rulings by federal appeals courts.
If the court denies review, the lower court ruling being appealed then becomes the final decision in that case.
In the Proposition 8 dispute, the 9th Circuit ruling would become the final decision and gay and lesbian marriages in California could resume. For the time being, that ruling has been put on hold, leaving Proposition 8 in effect, until the Supreme Court acts.
If the Supreme Court grants review, its written decision on the constitutionality of Proposition 8 would be expected by June 2013.
In its February opinion, a 9th Circuit panel said by a 2-1 vote that the ban violated the U.S. constitutional guarantee of equal treatment because of the specific circumstances of California marriage laws.
The panel said that because same-sex marriage was legal in California for several months in 2008, it was unfair to deprive gays and lesbians of that existing right without a legitimate justification.
"Because the 9th Circuit's decision is necessarily bound up with the particular circumstances presented by Proposition 8, that decision does not conflict with any decision of this court or any other court of appeals," the two same-sex couples argued to the high court on Friday.
Gay and lesbian marriage was briefly legal in California as a result of a May 2008 state Supreme Court ruling, which was nullified when voters approved Proposition 8 as a state constitutional amendment in November of that year.
In their appeal to the U.S. high court last month, Proposition 8's sponsors argued that whether marriage should be redefined to include same-sex couples is "a profoundly important question" and "a matter of great debate in our nation."
The sponsors contended that individual states, acting in the democratic process through legislatures or voters, should be allowed to define marriage within their state.