Former hedge fund executive David McCormick said he could not make up the wafer-thin deficit in the recount from last month's cliff-hanger vote.
Fewer than 1,000 ballots out of 1.34m votes cast separated the two rivals.
Mr Oz will now take on the Democratic candidate John Fetterman, who is recovering from a stroke.
Mr McCormick and Mr Oz fought each other to a standstill, the vote margin between them falling to within a 0.5% automatic recount threshold.
Mr Oz, a TV heart surgeon best known for his appearances on the Oprah Winfrey Show, released a video before the recount even began, describing himself as the "presumptive Republican" nominee.
He had been urged by his political patron, former President Donald Trump, to declare victory pending the official outcome.
Mr Oz would be the nation's first Muslim senator if elected.
Mr McCormick said in a statement on Friday: "Today I called Mehmet Oz to congratulate him on his victory."
He said he would extend his "full support" to Mr Oz.
In a series of tweets, Mr Oz thanked Mr McCormick for his "gracious" statement.
"Now that our primary is over, we will make sure that this US Senate seat does not fall into the hands of the radical left, led by John Fetterman," he added.
The contest in the key presidential swing state will be decided in the US mid-term elections in November.
The mid-terms fall halfway through a president's term in office. They decide who controls the two chambers which make up Congress - the Senate and the House of Representatives.
Following Mr McCormick's concession, the Democratic Senatorial Campaign Committee fired off a salvo at Mr Oz, labelling him as "a fraud and a scam artist" and a "self-serving millionaire".
Also on Friday, the Democratic candidate for the Senate seat said he "almost died" because of his stroke last month.
Mr Fetterman, who is Pennsylvania's lieutenant governor, said he should have been taking medication that a doctor prescribed for him in 2017.
His campaign also released a letter from his physician stating that the candidate has a disease called cardiomyopathy, which makes it harder for the heart to pump blood around the body.
The doctor said Mr Fetterman would be "fine" if he took his medication, improved his diet and exercised.
The Harvard-educated former mayor, who sported hoodie sweatshirts instead of suits on the campaign trail, is running as a left-wing Democrat.
A controversy that dogged him in his campaign is likely to come up during the general election.
In 2013, during Mr Fetterman's second term as mayor of Braddock, a town near Pittsburgh, he pursued an innocent black jogger who he wrongly thought had been firing a gun near his home.
Mr Fetterman, who is a hulking 6ft 8in and was armed with a shotgun during the confrontation, has refused to apologise for the incident.