The 56-year-old, who became chief executive of the stock vehicle racing organisation in 2003, was held overnight and released on Monday morning.
NASCAR issued an earlier statement Monday, before it announced France's leave of absence.
According to the police report, France was pulled over after failing to stop at a stop sign. He also failed a field sobriety test.
Tests showed France's blood alcohol content was 0.18.
Jim France, the son of NASCAR founder William HG France, also has an interest in the IMSA SportsCar Championship as its president. He is charged with aggravated driving while intoxicated and seventh-degree criminal possession of a controlled substance.
France was arraigned Monday at Sag Harbor Village Justice Court after spending the night in jail.
Sources told TMZ that throughout the process, he brought up his relationship to the Trumps.
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Felix Sabates, who owns a piece of Chip Ganassi Racing, told The Associated Press that Brian France gets unfairly criticized for the sport's recent struggles and that this is a time when they need to support him.
NASCAR says it takes France's arrest 'as a serious matter and will issue a statement after we have all of the facts.
Brian France has been NASCAR chief since 2003, following in the footsteps of his late father, Bill France Jr., and grandfather, Bill France Sr., who founded the organization in 1948.
Because NASCAR is privately owned and run in large part by Brian France, Jim France and Lesa France Kennedy, there is no clear indication of how hands-on Brian France was or what kind of void his absence will create.
Last month, France was forced to refute rumors that he was planning to sell the business.
So it is basically business as usual for NASCAR, and that's exactly the way it should be, said longtime France family adversary Bruton Smith. Aside from an occasional random call to SiriusXM NASCAR Radio, Brian France has said nearly nothing about NASCAR's woes this season. There's nothing to report on that. "Rumors are always interesting, but they're seldom right".
NASCAR's television ratings are down about 20 percent over the first 16 events of this season, according to figures from International Speedway Corp., which owns and operates the circuit.