The unbeaten American survived a brutal onslaught in the seventh round on his way to a 40th career victory and seventh title defence at New York's Barclays Centre.
Sky Bet price him up at 7/4 to beat Joshua should the pair meet this year, the British star is 2/5 and the draw 25/1.
IBF and WBA champion Joshua faces New Zealand's WBO champion Joseph Parker in Cardiff on March 31 and Wilder said he was fed up of just talking about the prospect of facing the Briton.
The 32-year-old said at his post-fight press conference: "How many times do I have to call him out? I've done enough, I've spoken enough. I'm the baddest man on the planet and whenever they're ready I'm going to be ready. My goal and my mission is to unify the division. I want to prove to the world I am the best."
Wilder also claimed his victory over the previously unbeaten Ortiz was more impressive than Joshua's defeat of Wladimir Klitschko.
"Wladimir wasn't the king, he was already dethroned," he said. "And really Joshua didn't win that fight, Wladimir lost."
Wilder certainly had to dig deep to see off Ortiz.
The challenger looked to be in control in the opening round, twice catching the champion with his left hand, but Wilder responded with a jab in the second before being forced into a corner.
From there, the crowd began to grow restless and jeered as Wilder struggled to land his punches in what was a slow start for both fighters.
By the fifth, Wilder finally found Ortiz with a right hand across the face which sent the Cuban to the canvas just before the end of the round.
But the seventh round almost proved to be the turning point as Ortiz retaliated with massive left hands, putting his opponent on the ropes and continuing the attack into the eighth.
"He had me in a whirlwind and I had to get out of there," Wilder said afterwards.
The late thriller continued into the next rounds when a knockout seemed to be Wilder's only chance of taking the bout, and a huge right uppercut ensured he produced it in the 10th, sending Ortiz to the floor as the referee waved for the fight to stop.
