The Illinois State Board of Elections rejected a complaint on Tuesday that sought to disqualify former President Donald J. Trump from the state’s primary ballot.

The appointed eight-member board determined unanimously that it did not have the authority to decide whether Mr. Trump had engaged in insurrection, the basis for the complaint. Its ruling can be appealed to the courts.

The board had appointed a former Republican judge, Clark Erickson, to hear arguments in the case. In an opinion made public over the weekend, Mr. Erickson said that he believed Mr. Trump had engaged in insurrection by attempting to remain in office after losing the 2020 election. But Mr. Erickson said he did not believe that the board had the authority to disqualify Mr. Trump on those grounds, and that the question should instead be left to the courts.

Mr. Trump, the leading Republican candidate for president, has faced official challenges to his candidacy in 35 states and has so far been found ineligible for primaries in two of them, Colorado and Maine. Mr. Trump is still likely to appear on the primary ballots in both of those states, because the ineligibility decisions are on hold while the U.S. Supreme Court considers an appeal of the Colorado ruling.

The Illinois challenge, like those in other states, is based on a clause of the 14th Amendment to the U.S. Constitution that bars government officials who “engaged in insurrection or rebellion” from holding office.

At a hearing last week in downtown Chicago, lawyers for Illinois residents objecting to Mr. Trump’s candidacy accused the former president of insurrection. They played video footage showing the riot by Trump supporters at the U.S. Capitol on Jan. 6, 2021.