Bowyer rushed to Newman's car after the cool-down lap, frantically trying to land blows through the window while Newman had his helmet on before crew members separated them.
The two had tangled on the track several times and were both angry and frustrated when NASCAR's prime-time spectacle, which for more than three decades has showcased the series' very best, ended.
Bowyer was upset that Newman turned him in Turn 4 late in the race.
"Where I come from," Bowyer said, "you get punched in the nose for that, and that's what he got."
Newman laughed off his rival's actions, mocking him for throwing punches at someone in a racing helmet.
"I think he should be embarrassed for himself," Newman said.
Look for Round 2 at Charlotte Motor Speedway in the Coca-Cola 600 next week.
Both drivers were called to the NASCAR hauler to talk with series officials.
Scott Miller, NASCAR senior vice president of competition, said officials will keep an eye on the two but believes the drivers will keep their issues in check going forward. Miller did not think that penalties would be assessed in the dispute.
"We're going to keep an eye on it," Miller said. "We certainly think we're in a good place with them, but we'll certainly have to monitor it moving forward with those two."
Bowyer started from the pole and finished 12th. He was a spot ahead of Newman in the 19-car field.
Newman is no stranger to NASCAR feuding. He has fought with Tony Stewart and both Busch brothers, Kurt and Kyle, during his career in racing. Bowyer was bothered by Newman earlier this season during qualifying at Texas, where he felt Newman had "clogged" the middle lane and affected Bowyer's qualifying.
Miller said NASCAR met separately with Bowyer and Newman. "They passed in the hall," he said.
Miller was asked about a specific plan to prevent another dust-up between the drivers.
"NASCAR's plan is to do just that -- watch them next week," he said. "If there are any shenanigans, we will intervene."
Miller would not go into the reasoning behind not issuing penalties.
"Every situation is different," he said. "We handle every one different. Depends on what they say. Depends on where we think they are, and that's between us and them."