Manager Kirk Gibson said an MRI exam Wednesday in Phoenix showed a tear in Hudson's ulnar collateral ligament. Tommy John ligament-replacement surgery is a possibility, but Gibson said Hudson could seek a second opinion.
"He just got the news. Let him sort it out," Gibson said.
Hudson left Tuesday night's game with the injury and revealed for the first time he had discomfort in the elbow earlier this season.
"You can't fault the kid for wanting to go out and pitch for his team," Gibson said.
Hudson said on his Twitter feed, DHuddy41, players are accustomed to trying to play through pain.
"Very rarely do guys go through a season and not feel some soreness or pain somewhere," Hudson said, adding "... it's hard to tell the difference between soreness and pain. Ask anyone in this game and they will all tell you the same thing. Put me out there."
Hudson was 16-12 with a 3.49 ERA for Arizona in 2011, but has had difficulty this season. He was on the disabled list from April 21 to May 27 with a right shoulder impingement and now faces a more serious injury.
He was removed after giving up five runs and seven hits in 1 2-3 innings on Tuesday night, leaving his ERA at 7.35. Hudson also struggled in his previous start, giving up seven runs in four innings in a loss to Seattle last week.
The Diamondbacks recalled left-hander Patrick Corbin from Triple-A Reno to pitch in long relief.
Gibson said Josh Collmenter will take Hudson's spot in the rotation Sunday at Milwaukee.
Hudson is the second Arizona starting pitcher to land on the DL in less than a week. Left-hander Joe Saunders was placed on the 15-day disabled list Saturday with a strained left shoulder.
Gibson said Saunders hasn't started throwing but is feeling better.
Hope for a boost for the rotation arrives on Thursday. Top prospect Trevor Bauer, the No. 3 pick in last year's amateur draft, will make his debut against the Braves in Thursday's series finale.
Bauer, 21, will be called up from Triple-A Reno, where he is 4-0 with a 2.82 ERA in eight starts.
The Diamondbacks also activated shortstop Stephen Drew from the disabled list to make his season debut Wednesday night against the Braves. Drew returned from right ankle surgery and went 1 for 4 in a 6-4 loss.
Right-hander Jonathan Albaladejo was optioned to Reno.