"Election is not war, and should never be seen as a do or die affair," President Muhammadu Buhari said on Wednesday.
Mr Buhari of the All Progressives Congress (APC) defeated Mr Abubakar of the opposition Peoples Democratic Party (PDP) in Saturday's presidential election.
The president scored 15,191,847, while Mr Abubakar polled 11,262,978.
Mr Buhari was declared winner early on Wednesday by the chairman of Independent National Electoral Commission (INEC) Mahmood Yakubu.
But Mr Abubakar and his party rejected the outcome of the election and said they will seek legal redress.
But speaking shortly after receiving his certificate of return from the INEC chairman, Wednesday, Mr Buhari called on his opponent to accept defeat.
"Election is not war, and should never be seen as a do or die affair. I pray that we all accept this democratic approach to elections, however contentious," he said.
The president said now that the elections are over, and a winner declared, "we must all see it as a victory for Nigeria, our dear country".
"That was why I encouraged my teeming supporters, in a speech I read earlier today, not to gloat. Our God-given victory is enough cause for joy, without deriding those who were in the opposition. All Nigerians, going forward, must stand in brotherhood, for a bright and fulfilling future.
" I, therefore, want to assure that we will continue to engage all parties that have the best interest of Nigerians at heart," he said.
Mr Buhari said his government will remain inclusive and that all doors will remain open. That, he said, "is the way to build the country of our dream; safe, secure, prosperous, and free of impunity and primitive accumulation by those entrusted with public offices".
The president congratulated all the presidential candidates and their teams on a hard-fought campaign.
"We may have had different views during the campaign, but the one thing most of us have in common is love of our country and our desire to improve conditions for Nigerians".
He also said, "from the comments of several observers both local and foreign, it is obvious that the elections were both free and fair".
Mr Buhari said for him and the government he runs, the hard work to deliver a better Nigeria continues, building on the foundations of peace, rule of law and opportunities for all.
"We will roll up our sleeves afresh, and give it our all. We have no other motive than to serve Nigeria with our hearts and might, and build a nation which we and generations to come can be proud of," he said.
Mr Buhari thanked the over 15 million citizens who voted for him. He also thanked those who contributed to the running of his successful campaign.
"I can assure that you will see a country moving to the Next Level, as we consolidate on our fundamental areas of securing the country, reviving the economy, and fighting corruption.
"Many other focal areas, as stated in our campaign manual, will be added, to the good things we plan for our country," he said.
Mr Buhari also paid tribute to those who lost their lives either through accidents or mindless violence during the course of the elections.
"My heart and thoughts are with the families of the victims. May God comfort and give them the fortitude to bear the loss," he said.