"As one of the largest economies in the world and home to one in nine Americans eligible to vote, California has a special responsibility to require this information of presidential and gubernatorial candidates", Democratic Governor Gavin Newsom wrote in his statement that accompanied his signature on the bill.
"The Constitution is clear on the qualifications for someone to serve as president and states can not add additional requirements on their own", Murtaugh said in a statement.
Trump's campaign staff said state laws on presidential candidates' eligibility were unconstitutional.
Trump campaign spokesman Tim Murtaugh said there were "good reasons" Brown vetoed the measure.
Tax returns show income, charitable giving and business dealings, all of which Democratic state lawmakers say voters are entitled to know about.
While the law is expected to be challenged, Trump could sidestep its requirement by not competing in the California primary.
The bill, known as the "Presidential Tax Transparency and Accountability Act", would also apply to gubernatorial candidates and requires the release of tax returns for the most recent five years. Five years of health records?
While states have authority over how candidates can access the ballot, the U.S. Constitution lays out a limited set of qualifications someone needs to meet to run for president, said Rick Hasen, a professor specializing in election law at the University of California-Irvine School of Law.
In 1994, the Supreme Court ruled that states - in this case, Arkansas - could not add on their own qualifications for congressional candidates. Newsom said his state is "well within its constitutional right", but critics of the tax-return disclosure effort say the Constitution sets out requirements to be president and states can not add new ones.
In this December 22, 2017, file photo, President Donald Trump speaks with reporters after signing a tax bill and resolution in the Oval Office of the White House in Washington.
The president's personal attorney, Jay Sekulow, responded to the passing of the California law, saying Trump expects to challenge it in court.
In a tweet to Trump, McGuire said: "If you want to be on the CA primary ballot, release your returns".
"The Supreme Court might, if faced with the issue, hold that state legislators can not require tax returns of presidential candidates even given state legislatures' much greater power over presidential elections", he wrote at the time.
It was passed by California's heavily Democratic legislature last month, and appears only to affect candidates running in their parties' primary votes, not the November 2020 poll itself.
Treasury Secretary Steven Mnuchin has rejected requests by the U.S. House of Representatives' Ways and Means Committee to the Internal Revenue Service to turn over six years of Trump's returns.