There is no data to support this assertion. Pitchers commonly train with weighted balls, if that was an injury risk they wouldn't. The ball weighs 5 ounces. My suggestion is increasing that to 5.5 ounces. A football weighs 15 ounces and NFL quarterbacks have a lower rate of throwing injuries than MLB pitchers.
A pitcher would probably like a ball that is lighter and has a better grip because he can get more spin and strike out batters more easily. Pitchers WILL NOT like a heavier ball that spins less. The idea isn't to make pitchers happy, it's to get us out of the "Home Run Derby" era of baseball and get more athletic action into the game. A lighter ball that spins more would mean even more strikeouts, even fewer balls in play and an even more boring MLB.