Why did the character of Steve leave?

Gary Clarke (Steve) made the mistake of supporting James Drury in a dispute with the executive producer, Frank Price, about which horse Drury should ride. In the middle of second season Drury decided not to ride the white horse (named Joe) choosing to switch to a paint horse.

Price insisted Drury ride Joe because changing horses meant re-filming lead-in and fade shots (remember the last shot of every show of Virginian sitting on Joe). Drury became difficult over scene setups, dialogue and other small but important issues and McClure and Clarke supported Drury to make life difficult for Price. Price knew McClure was too important to the series to fire but that did not apply to Gary Clarke.

Clarke's misfortune was being the third cowboy, on the show. Clarke was first dropped from regular to recurring guest star, in third season and his contract was not renewed for fourth season. Drury got the message and began riding Joe (which took McClure off the hook).

Drury developed a real affection for Joe and visited the horse until he died in the 1980s.

There is a book about the show named "History of TV's The Virginian" by Paul Green that provides a year by year description of the filming of the show. All of the major actors gave in-depth interviews to Green along with Frank Price. If you like the TV show, it is a must read.