In his petition, Abubakar of the Peoples Democratic Party (PDP) claims the election was invalid because the Independent National Electoral Commission or INEC failed to comply with election law by not uploading the results.
Among other charges, the petition claims manipulation of votes and that Tinubu did not meet constitutional requirements to be declared the winner.
It calls for Abubakar to be named the winner or for fresh elections to be held.
Abubakar's petition filed late Tuesday in an Abuja appeals court follows one from the Labor Party candidate Peter Obi, who came third, as part of what will likely be a prolonged legal battle over the February 25 election.
Former Lagos governor and ruling All Progressives Congress (APC) party strongman Bola Tinubu won the election with around 37 percent of vote, but opposition parties charge delays in electronic uploading of results aided massive ballot rigging.
The constitution allows the courts 180 days to make a ruling on election matters and plaintiffs can then appeal to the supreme court.
Abubakar, who was on his sixth bid for the presidency, also challenged the results in 2019 when he lost against incumbent President Muhammadu Buhari.
The supreme court dismissed his fraud claims months later.
- Voting turmoil -
INEC has denied any illegal activity, though acknowledged technical problems with the so-called BVAS voting machines.
Nigerians had hoped the presidential ballot would give them a chance to be heard, but many were disappointed by the way the election was conducted.
Voters and opposition parties complained last month that technical mishaps with voting machines caused delays and allowed for vote rigging, which the electoral commission has denied.
International observers, including from the European Union, noted major logistical problems, disenfranchised voters and a lack of transparency.
Earlier on Tuesday, president-elect Tinubu appealed for unity, saying "the time for politicking is gone".
Nigeria's ruling APC party also won the majority of governorships contested in last weekend's local elections, results showed, following a ballot marred by voter suppression and violent intimidation which the US government called deeply troubling.
