Gantz's centrist Blue and White party is slightly ahead of Netanyahu's Likud in the second round of voting this year, which was held Tuesday. However, they both lack the numbers of seats in the 120-member parliament to form a ruling bloc.
Netanyahu, Israel's longest-serving leader, urged Gantz, the former military chief of the regime, to meet him Thursday, because he had pledged to form a right-wing government led by Likud.
"But to my regret, the election results show that this is impossible," Netanyahu said in a video message on Thursday. "Benny, we must set up a broad unity government, as soon as today."
Gantz, in response, said without mentioning Netanyahu and his request that he himself was going to form a "liberal" coalition, implying that he would not become allies with Netanyahu.
Moshe Yaalon, another top leader of the Blue and White, gave a more detailed response later in the day, saying his party was not willing to join forces with Likud mainly because of the looming corruption charges against Netanyahu.
"We will not enter a coalition led by Netanyahu," Yaalon said.
"The time has come for you to tell Netanyahu: 'Thank you for all you've done'," Yaalon urged Likud members, who are yet to show signs of rebellion against the PM.
The swift rejection left Netanyahu "surprised and disappointed," prompting him to repeat the call.
"It's what the public expects of us," Netanyahu said later in the day.
Gantz had taken a similar position over the course of the campaign and suggested several times that an alliance would only be possible if Likud dumped Netanyahu.
The election came five months after an inconclusive vote in August, where Netanyahu once again failed to form a majority government.
With 99 percent of votes counted, Israeli media are saying that a Likud-led right-wing bloc looks set to control 55 of the parliament's 120 seats, with up to 57 for a center-left alliance.
Despite their rivalry, Netanyahu and Gantz had only narrow differences on key issues during their campaigns, meaning that even if Netanyahu was to stand down after five terms in office, significant changes in the Tel Aviv regime's policies on relations with the United States, the standoff with Iran or the Palestinian conflict would most likely remain unchanged.
Once all the votes are counted, Israeli President Reuven Rivlin will meet with the parties that won seats and will ultimately give one party leader up to 42 days to form a government.
The deadlock has turned the regime's former minister of military affairs Avigdor Lieberman, who is now head of the far-right Yisrael Beitenu party which has eight seats, as a key influence in the coalition-building process.
Lieberman remained open to the idea of a secular unity government but said Thursday that he was unconvinced by Netanyahu's unity offer.
In a statement, he called Netanyahu's offer "no more than a trick and an attempt to paint a false picture to prepare public opinion for a third round of elections."
Lieberman resigned from Netanyahu's cabinet earlier this year and has become one of his firm critics ever since.