The veteran player will now miss his team's opener against the Tennessee Titans, but Bowe has vowed to help them push towards the Super Bowl once he returns.
"I am extremely sorry that I can't be on the field for the first game of the 2014 season because I made an error in judgment," he said in a statement.
"I take responsibility for my mistake, and it will not happen again.
"I hope that my teammates, coaches, and Chiefs fans accept my sincere apology. I will be supporting my teammates in week one, and I cannot wait to get back on the field to help us win in week two and beyond as we work towards our goal of winning the Super Bowl."
The new NFL season gets underway in September.