Wilder, 32, floored the previously unbeaten Cuban twice before knocking him out in the 10th round in New York.
IBF and WBA champion Briton Joshua, 28, will fight New Zealand’s WBO champion Joseph Parker in Cardiff on 31 March.
“I want to unify. I am ready whenever those guys are,” said Wilder.
He survived a scare in the seventh round to claim the 40th win of his unbeaten career following a right uppercut at the Barclays Center.
“I figured out a way to come back. A true champion always finds a way. That’s what I did tonight,” said Wilder, who has successfully defended his title seven times.
Joshua’s promoter, Eddie Hearn, said that if he beats Parker a unification fight with Wilder is likely to happen this year.
“It’s the biggest fight in world boxing,” Hearn told Sportsweek on BBC Radio 5 live.
“He has the biggest test of his career against Parker but if he wins it Anthony wants to fight Wilder.
“He thinks he can beat him and Wilder is beatable, but Wilder is also very dangerous.
“Both Wilder and Joshua are electric to watch, both punch very hard and both have shown signs of vulnerability.
“You have to give the fans what they want, otherwise over time they lose interest. I think it will happen 2018. I don’t think there is any holding AJ back.”
Wilder knocked Ortiz down in the fifth round, but the 38-year-old responded with a right hook and left hand and a barrage of his own as Wilder just managed to survive the seventh.
“He hit me with flurry of punches but it wasn’t hard,” said Wilder. “He knocked me off balance. I just had to get my fundamentals back and I did.”
Ortiz, who had won all 28 of his previous fights and weighted in almost 30lb heavier than Wilder, was knocked down again in the 10th before the referee stopped the fight with 55 seconds remaining in the round.
Sergey Kovalev retained his WBO light-heavyweight title by beating fellow Russian Igor Mikhalkin at Madison Square Garden.
The ring doctor stopped the fight in the seventh round as a dominant Kovalev, 34, claimed his second successive knockout since losing his three titles to Andre Ward in 2016.
“I am back,” said Kovalev. “He looked easy but he was a good boxer and had good stamina.”
On the same card, Russia’s Dmitry Bivol, 27, stopped Cuba’s Sullivan Barrera in the 12th round to retain his WBA light-heavyweight title. (Courtesy BBC)