Public accommodation not private
That depends upon the state and courts.
the Supreme Court in Lloyd Corp. v. Tanner held that private property generally does not lose its private character merely because the public is invited to use it for designated purposes. Subsequent cases confirmed that the First Amendment does not protect free speech in privately owned malls.
State Interpretations Vary
the Arizona Court of Appeals held that the free-speech and initiative provisions of the Arizona Constitution did not protect the solicitation of signatures in privately owned malls for the recall of the then-governor in Fiesta Mall Venture v. Mecham Recall Comm. in 1988.
courts — including those in Arizona, Georgia, New York, Pennsylvania, and South Carolina —...in these states also have noted that state funding of private development does not constitute sufficient state involvement to make the state constitution protections of free speech applicable.
Massachusetts limits protection to ballot access; Oregon limits protection to initiative petitions; and New Jersey limits protection to leafleting and related speech supporting or opposing causes, candidates, or parties.
In a 1979 decision, the California Supreme Court held that free speech was protected under the California Constitution even on privately owned property and without the involvement of state action in Robins v. Pruneyard Shopping Center. - https://www.ccim.com/cire-magazine/articles/states-speak-out-free-speech-malls/
Connecticut Supreme Court rules malls can block free speech....In a seminal ruling 20 years ago, the state Supreme Court declared that shopping malls – although teeming with members of the public – are private properties that can ban political and other free speech activities from their premises. And the high court is holding fast to the tenets of that ruling. - https://aldeilis.net/english/shopping-malls-can-ban-free-speech/
Free speech in malls; [NJ] State Supreme Court rules against restrictions...Last week, by a unanimous decision, the state Supreme Court agreed, establishing one more huge precedent for opening the malls to public discourse. ...“Throughout much of New Jersey today there is no place to go, other than shopping centers and regional malls, if one is to have an opportunity to meet face-to-face with large groups of people,” wrote Justice Daniel O’Hern. “Except for those commercial centers, the public common has largely ceased to exist.”...
Gave something to both sides In its ruling last week, the state Supreme Court unanimously reaffirmed the free speech rights established by the 1994 ruling and said the malls had to be reasonable in setting up insurance demands for groups seeking to gather signatures or pass out pamphlets. The court ruled that malls must take into account the past of such groups, and must provide reasonable access if the group has no history of violence or disruption
The ruling cannot be appealed to the U.S. Supreme Court because it is based on state law. Federal courts have already determined that the U.S. Constitution does not protect free speech in shopping malls – despite a 1924 federal ruling that leafleting is a basic right of American citizens. ...” Malls can also prohibit pickets, bullhorns, megaphone placards, and demonstrations. The ruling also does not apply to highway strip malls, football stadiums, or theaters, since these locations do not approach the multitude of uses found at regional shopping centers. But the ruling also gave malls something as well since the state Supreme Court overturned part of a previous lower court ruling that would have provided even more access to the malls. - https://archive.hudsonreporter.com/2000/06/23/free-speech-in-malls-state-supreme-court-rules-against-restrictions-2/