The altercation happened after Paul was called for a foul with 5:15 left in the third quarter of the Spurs' 100-75 loss to New Orleans on Wednesday.
Besides his streak, Bowen will lose about $37,000.
"The league is trying to send him a message more than anything," Spurs coach Gregg Popovich said. "The incident itself, I've watched a lot of times, and there was no real contact. It doesn't reach the level of other suspensions that I've seen, either in physicality or harm."
Bowen has built a reputation as one of the NBA's best defenders and perhaps its most physical on the perimeter.
Popovich said the suspension was a tough way for Bowen's streak to end.
"I'm sure he's very disappointed," Popovich said. "He was very proud of that. He takes great pride in taking care of himself, especially at his age."
The 36-year-old Bowen is averaging 5.8 points and 2.7 rebounds this season. He was on the NBA All-Defensive team the previous seven seasons and was runner-up for Defensive Player of the Year the last three seasons.