A military convoy took the coffin, draped with a Palestinian flag and yellow flowers, to Darwish's grave near the Palace of Culture, about 4km (2.5 miles) away from central Ramallah.
"Oh Mahmoud, Oh Mahmoud, you rest and we will continue the struggle," mourners chanted as the poet was buried to a 21-shot salute.
His poems were recited over loudspeakers, and Mr Abbas read a eulogy.
"He was the master of the word and wisdom, the symbol who expressed our national feeling, our human constitution, our declaration of independence," Mr Abbas said.
"You remain with us, Mahmoud, because you represent everything that unites us."
Darwish's body was flown back from the US to Amman, Jordan, on Wednesday where an honour guard saluted as Palestinian Liberation Army officers carried the flag-draped coffin from the plane.
The coffin was then taken by military helicopter to Ramallah for the funeral - the biggest in the West Bank since that of Yasser Arafat in 2004.
The BBC's Aleem Maqbool, in Ramallah, says people of all backgrounds in the West Bank feel they had a personal connection to the poet, and take pride in a man who told their story in a way they could not.
Darwish was regarded as a national icon, whose work was often based on his experiences of life in exile.
"He symbolises the Palestinian memory," one Palestinian mourner told the BBC.
"He intended to convey a message: in the end we are all human beings and we have to work collectively for the sake of humanity."
Thousands would flock to his recitals and his poems were transformed into popular songs and used in political speeches. The words he wrote now form part of Palestinian daily life, our correspondent says.
Darwish won many international prizes for his work
Nor was he shy of talking of his people's shortcomings.
Darwish penned fierce criticism of the divisions among Palestinians, believing, in some ways, what they were doing to themselves was worse than anything others had done to them.
He also wrote the famous speech Arafat delivered at the United Nations in 1974: "Today I have come bearing an olive branch and a freedom-fighter's gun. Do not let the olive branch fall from my hand."
There is little doubt his work, not just on the Palestinian cause, but on love and hope and death, will endure across the Arab world, our correspondent says.
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