Shari Redstone jockeyed for years to succeed her father atop a media empire that includes CBS Corp. and Viacom Inc. Having reached that summit, the newly minted media mogul believes merging those companies is the best way to preserve her family’s legacy and wealth.

Leslie Moonves, a former actor and TV programming wizard, has a record as CBS’s chief executive that has earned him a reservoir of support on Wall Street. He believes the deal would be a huge mistake and could tarnish his legacy late in his career.

The...