White was arrested on Saturday while in Arizona at spring training. He spoke with team management Sunday morning.
The Rockies say White understands he made a mistake and the embarrassment it has caused.
The 23-year-old White was acquired last summer from Cleveland in the trade for Ubaldo Jimenez.
White is vying for a spot in the Colorado rotation. He went 2-4 with an 8.42 ERA in seven starts with the Rockies last season.