The next round of global climate change negotiations is scheduled to take place in the Mexican resort town, Cancun, in November.
After the failure climate change talks in Copenhagen, Denmark, in December 2009, the Cancun summit will be crucial if a global agreement is to be reached.
"We need to be more careful and serious about what the expectations are," Espinosa said.
There is a near universal consensus among the world's leading scientists that human activities, such as burning fossil fuels, are causing the planet to heat up.
While Mexico played down hopes for an agreement on climate change, others celebrated World Environment Day with species conservation in mind.
In Rwanda, film stars and United Nations officials gathered for a ceremony to name 11 endangered baby mountain gorillas.
Internet users from around the world chose the name "Zoya" as the most popular for the gorillas, Don Cheadle, the American actor, said.
World Environment Day raised more than $85,000 for gorilla conservation, the United Nations Environment Programme said in a statement.
"From frogs to gorillas, from huge plants to tiny insects, thousands of species are in jeopardy," Ban Ki-moon, the UN secretary general, said in a statement.
Dmitry Medvedev, the Russian president, called for the world's leading economies to create a fund to insure against large-scale environmental disasters like the Gulf of Mexico oil spill.
The first World Environment Day was celebrated on June 5, 1972.