Woodward, who guided the Exiles into the top flight in only his second year in the job this season, stormed out of the club's annual general meeting and then branded them as "probably illegal".
The double British Lion, a member of England's 1980 Grand Slam team, said: "They passed a resolution that said that the people working on the general committee would 'normally' be of Irish descent.
"They were prepared to offer dispensations for Englishmen like me, but I did not want to be regarded as a special case. When I joined the club a couple of years ago, I was absolutely assured that it was an open club.
"The vote to retain the Irishness on a formal basis was narrow, but it did show a bigoted element among the members and that Englishmen could be there only on sufferance.
The London Irish chairman, Duncan Leopold, promised the club would make every effort to persuade Woodward to change his mind.
"We believe this is a misinterpretation by him of our position," Leopold said.Reuse content