The #BringBackOurGirls group should be ashamed of manipulating "the victims of terrorism", he said.
Mr Jonathan had been due to hold his first meeting with some of the girls' parents on Tuesday.
Islamist group Boko Haram captured more than 200 girls during a raid on their boarding school in Chibok in April.
Mr Jonathan had been widely criticised for failing to meet distraught parents and not doing enough to rescue the girls.
#BringBackOurGirls was a global campaign launched on social media to secure the release of the girls.
Mr Jonathan agreed to the meeting on Monday following an appeal by Pakistani rights campaigner Malala Yousafzai.
Mr Jonathan had planned to meet 12 parents and five girls who escaped shortly after being captured but the parents withdrew at the last minute without giving any reason.
"Unfortunately, political forces within the Nigerian chapter of Bring Back Our Girls have decided to take this opportunity to play politics with the situation and the grief of the parents and the girls. They should be ashamed of their actions," he said in a statement.
"Those who would manipulate the victims of terrorism for their own benefit are engaging in a similar kind of evil: Psychological terrorism," he added.
Mr Jonathan's spokesman Doyin Okupe said the president was "extremely distraught".
"He cannot understand this. He cannot fathom this - that Nigeria's mothers will do this type of thing for children," Mr Okupe said.
Nigeria is due to hold presidential elections next year.
Mr Jonathan has not yet declared his candidature, but speculation is rife that he intends to run for another term.
Last year, he declared a state of emergency in the north-eastern states of Borno, Adamawa and Yobe, which are worst affected by the insurgency.
All three are under opposition control.
New York-based Human Rights Watch says more than 2,000 civilians have been killed in Nigeria this year by Boko Haram.
The deaths occurred in around 95 separate attacks in more than 70 towns and villages in the north-east, where Boko Haram launched its insurgency in 2009.
Nigeria's president has accused activists of "playing politics" after his meeting with parents of the abducted schoolgirls was called off.