Other reports from Gaza say Johnston is healthy and negotiations to secure his release are under way with the abductors.
Abbas and Ismail Haniya, the Palestinian prime minister, have condemned the kidnapping and have ordered security forces to secure his release as quickly as possible while Haniya said he told security forces to do all they could to find him.
He also said that Palestinian journalists who came to the Palestinian territories were guests of the Palestinian people and should be protected.
Kidnapping were unacceptable and "harm the civilised face of our people," Haniya said.
Johnston has is one of the few Western reporters working out of the Gaza Strip and has been based there for the past three years.
Abductions of foreigners have become increasingly common in the impoverished Gaza Strip, with about 20 such cases in the past year.
Kidnappers normally use the hostages as bargaining chips to gain concessions from the Palestinian Authority, and they have all been released unharmed.
The last journalist snatched in Gaza City was Jaime Razuri, the Peruvian AFP photographer, who was seized on January 1 and released unharmed seven days later.
Home to 1.4 million people, the Gaza Strip has spent the last year wracked by lethal Palestinian infighting, a punishing Western aid boycott that has sent the economy into a tailspin and deadly Israeli military operations.