Lopez Obrador, who promised to govern for the country's poor, is demanding a recount in the July 2 election, from which Felipe Calderon of the conservative ruling National Action Party (PAN) party emerged with an advantage of less than 0.6%, or about 244,000 votes.
Although the protests have so far been peaceful, there are concerns violence could erupt as protesters grow frustrated.
Calderon appeared before Mexico's electoral tribunal on Sunday to argue that the election was clean and a full recount was unnecessary and illegal.
"The question is whether we Mexicans are going to resolve our differences with pressure tactics and marches, or with reason and by the law," Calderon said in his statement.
"The fundamental thing is that we have democracy in our country"
Andres Manuel Lopez Obrador, Mexico presidential candidate
However Lopez Obrador said his demand for a recount was not just about recognising his "victory" in the election.
"The fundamental thing is that we have democracy in our country," he said.
The judges have until August 31 to decide whether or not to order a recount.
European Union observers said they found no evidence of foul play in the election.
Opinion polls reportedly show that while slightly more than half the country thinks Calderon won, more than a third believe there was fraud and half want a recount.
Sunday's march was the third major protest in the last three weeks, and one of the biggest in Mexican history.
City police said up to two million people took part while federal police said there were less than 200,000.