Bukele, who took office June 1, said late Sunday that the drownings were 'a great tragedy.'
On Monday, about 200 relatives and friends followed a hearse bearing the bodies of Óscar Martínez and his 23-month-old daughter Angie Valeria Martínez inside La Bermeja municipal cemetery in the capital city of San Salvador.
The ceremony was private, and journalists were not allowed access.
Many wore black and wept. They carried flowers and green palms, and some held signs bearing the logo of the Alianza soccer team favored by Martínez, who belonged to a group that supports the club.
'We can speak blame to any other country, but what about our blame?' Bukele said in lengthy remarks, much of them delivered in English.
'I mean, what country did they flee? Did they (flee) the United States? They fled El Salvador. They fled our country. It is our fault.'
Óscar and Angie Valeria were found face down in shallow water on the Mexico side of the river across from Brownsville, Texas last Monday morning a day after they were swept by a rip current as the mother, Tania Vanessa Ávalos, 21, watched in horror.
Photos from the scene show his black shirt hiked up to his chest with the girl's head tucked inside. Her arm was draped around his neck suggesting she clung to him in her final moments.
The young father and his daughter were just the latest victims claimed by the humanitarian crisis that has affected residents from Central America and other neighboring countries in the Latin America region who abandoned their homes due to violence and poverty and died in their search for the American Dream.
Bukele charged that the Salvadoran government didn't do enough to protect the 25-year-old Salvadoran and the innocent toddler.
'We haven't been able to provide anything, not a decent job, not a decent school,' he said.
'What if there's a little girl who had a decent school here, a decent health care system for her and her family, a decent house with water supply, a job for his parents, for his mother and his dad, a decent job, living in a zone where a gang member would not come to rape her and kill her family?'
He said the onus is on his government to make the country a safer place - and one where migration is 'an option, not an obligation'. Bukele also indicated there's blame to be shared among governments.
U.S. policies designed to deter Central Americans and others from coming have stalled thousands on the Mexico side of the border as they wait to request asylum in the United States.
Family members said last week that Óscar had grown frustrated with the amount of time it was taking for officials in Matamoros, Mexico, to conduct their petition interview. The desperate migrant told his last Sunday it was time to make the trip across the Rio Grande and turn themselves in to U.S. border patrol officials before tragedy struck.
Mexico is stepping up immigration enforcement under intense pressure from the Trump administration.
And Bukele, who took office a month ago, said his own country - the smallest in Central America - shares responsibility.
Many of those trying to reach the U.S. border in recent months have said they were fleeing grinding poverty, a lack of opportunity and violence in the gang-dominated Northern Triangle region of El Salvador, Honduras and Guatemala, or a combination of these factors.
Rosa María Ramírez, Óscar's mother, said last week that the family had hoped to live and work in the United States for a few years, saving enough money to return and build a home of their own.
'These are situations that make visible the problem of migration, something that happens daily,' said Mario Vega, pastor of the Elim Protestant church, one of El Salvador's largest.
'Yesterday it became known that another Salvadoran died in a detention center in Texas, and this is something that happens every day.'
Vega criticized U.S. policies on migration and expressed hope that all countries would heed a U.N. call to respect migrants' human rights.
'May God will that this tragedy sensitize people,' he said.
Comment: The world's problems cannot be understood through black and white thinking, and the migrant crisis is proof of that. It is heartening to see El Salvador's president claim responsibility. More leaders need to do the same. The US bears responsibility here as well, as do many other countries who have participated in creating this mess.