The duo spent most of the race battling for eighth and swapped positions repeatedly before making contact at the final bend on the last lap as Hayden tried to pass down the inside.
As both Ducatis bounced over the high kerbs and onto the venue's oval, Tech 3 Yamaha rider Bradley Smith was able to accelerate past them and steal eighth.
Hayden and Dovizioso were called to race direction afterwards but no action was taken.
Despite highlighting Hayden's 'aggressive' riding, Dovizioso said he preferred not to stoke controversy.
"He wanted to get by me at any cost, using a really aggressive manoeuvre," said the Italian.
"I didn't expect it because I hadn't left the door open.
"Nicky came beside me and we touched, and it certainly wasn't a safe move, but we can say it was a racing incident.
"We were called to race direction because of course it's important to talk about these things and immediately clear them up.
"Honestly, I'm quite annoyed to have lost two positions, and I also wanted to keep Smith behind me.
"Still, there's no sense in creating controversy in a situation like this, so for me it's over."
Hayden admitted the clash was his fault.
"We went back and forth a lot, and then in the last corner on the last lap, we had contact, which was my fault," he said.
"When the door started to close, I was already committed and didn't want him to take out my front wheel."
Ducati announced last month that it will drop Hayden at the end of 2013 after five seasons. Cal Crutchlow will be Dovizioso's new team-mate.