But Mr. LaPierre, in a stinging letter sent on Thursday night to the N.R.A.’s board, accused Mr. North of threatening to leak damaging information about him and other N.R.A. executives unless he stepped down.

“Yesterday evening, I was forced to confront one of those defining choices — styled, in the parlance of extortionists — as an offer I couldn’t refuse,” Mr. LaPierre wrote. “I refused it.”

Even as the leadership tussled behind the scenes, President Trump addressed the N.R.A. faithful at the convention on Friday and proclaimed himself a champion of gun rights. In a speech that was part political rally and part pep talk, he said his administration would not ratify an arms treaty designed to regulate the international sale of conventional weapons.

The power struggle within the N.R.A. is an abrupt escalation of a legal battle between the organization and Ackerman McQueen. The Times reported earlier this year that prominent members of the N.R.A. board had grown dismayed at the performance of Ackerman because of its NRATV online media service, which has drifted into right-wing politics far beyond gun rights. Ackerman employs Mr. North, who hosts an NRATV series called “American Heroes.”

It is not clear whether Mr. North has the board support to oust Mr. LaPierre, who has led the N.R.A. for decades. Previously, the presidency has been a ceremonial position, though Mr. North, in documents reviewed by The Times, has asked for it to be a paid post. A key factor will be Chris Cox, who runs the N.R.A.’s Institute for Legislative Action and is effectively the group’s second-ranking official.