Matthews won a photo finish sprint to claim the 16th stage ahead of Edvald Boasson Hagen and Degenkolb.
But German Degenkolb was angered by the line Matthews took in the finish and waved his hand at the Aussie as they crossed the line.
He then allegedly grabbed Matthews around the neck in the area just beyond the finishing line.
"I was waiting for the result and he grabbed me on the way past," said the 26-year-old.
"The officials saw it. We'll see what the officials will do about it.
"I don't think it's very sportsmanlike.
"From my perspective, I don't think I did anything wrong. I didn't change my line, I sprinted in a straight line.
"I'm not sure what was wrong with Degenkolb, but that's up to him."
Matthews also claimed another 50 points in the sprinters' green jersey competition to close to 29 points behind Quick-Step's Marcel Kittel, who has won five stages this year.
Kittel had been dropped early in the stage on a third category climb as Matthews's Sunweb team set a fierce pace.
Crosswinds in the final 15km of the 165km stage from Le Puy-en-Velay to Romans-sur-Isere then scattered groups all over the road meaning a select group of less than 30 riders reached the finish together.
"We took it from the word go, attacking the whole climb to make it hard for Quick-Step to shut us down," said Matthews, who also won a stage on the Tour last year.
"I kept attacking on the first climb, I felt quite good and when I got to the top of climb I heard Kittel was distanced by almost a minute. I called the boys and they came to the front with big smiles on their faces and went full gas."
The crosswinds got rid of other top sprinters Andre Greipel and Alexander Kristoff, leaving little competition for Matthews at the finish.
"It was a great day, it went 10 times better than was planned," added Matthews.
'Nothing is lost'
"I can't say I'm happy at losing all those points today but what can I do," said Kittel.
"It wasn't our day today but we have to move on and think of the next chances not missed opportunities."
He added: "Nothing is lost."
Chris Froome retained the overall lead in a hectic finale baulked by crosswinds that saw top 10 contenders Daniel Martin, Louis Meintjes (both 51sec) and Alberto Contador (1min 33sec) lose time.
"Everyone knew that the last 20km would be in the wind with crosswinds," said Sky's Froome.
"It was there that we decided we had to go, to ride hard and try to make a selection."
Frenchman Romain Bardet was almost caught out but was paced back to the lead group by team-mate Oliver Naesen, while Fabio Aru had to bridge a small gap by himself.
The Italian remains second at 18sec with Bardet third at 23sec.
Froome's Spanish team-mate Mikel Landa reclaimed fifth place from Martin, who dropped to seventh, while Nairo Quintana climbed back into the top 10 at the expense of Contador.
Martin had picked up a few seconds in each of the previous four stages but lost more in this one than he had previously managed to garner.
"I was a bit sick yesterday and then some of the other guys have been sick as well," said Quick-Step's Martin.
"I wasn't in a bad position, but it was probably me who didn't have the legs or the power in the wind.
"It happens. It's unfortunate. We worked so hard to be in this position but it's not over."