"I regret any disruption my accident last weekend may have caused members of the community and want to thank the local authorities," Kidd tweeted.
He followed up with, "I'd also like to thank my family and friends for their support."
Kidd, 39, was charged with a misdemeanor after a single-car accident in the Hamptons. At around 2 a.m. Sunday, his 2010 Cadillac Escalade struck a telephone pole and went into the woods in Water Mill, a hamlet of the Town of Southampton, N.Y.
He was treated for minor injuries at Southampton Hospital.
The Southampton police told The New York Times that Kidd refused tests to measure the level of alcohol in his system. Kidd also declined a blood alcohol test at the hospital, according to the Times report.
The Times, citing paperwork filed by police for prosecutors, reported that Kidd told police, "I had a few drinks. I was going home from a club. I don't know what happened."
Kidd is scheduled to appear in court on Sept. 12. He has pleaded not guilty to a charge of driving while intoxicated.
He also was charged with failure to stop and refusing a breathalyzer at the scene of the crash. His lawyer, Edward D. Burke Jr., told the Times those charges were "traffic violations."
Three days before his accident, Kidd signed a free-agent contract with the Knicks after spending the previous 4½ seasons with the Dallas Mavericks. The 10-time All-Star has also played for the Phoenix Suns and New Jersey Nets, twice leading the Nets to the NBA Finals. He was a member of Dallas' championship team in 2010-11.