"It is high time the Palestinian people have a new leadership," he said.
But Meshaal's comments were not supported by all Hamas members, many of whom said the PLO should be reformed rather than replaced.
Hamas is not a member of the Palestinian umbrella group, which Egypt founded in 1964 and which Fatah took over in 1968.
Speaking in Cairo, Abbas said: "We say this with utmost clarity - no dialogue with whoever rejects the Palestine Liberation Organisation."
Abbas made his comments after visiting Palestinian casualties being cared for in a hospital in Cairo.
He also directly criticised Hamas, saying they had put the lives and aspirations of Palestinians at risk by promoting conflict.
"They ... have taken risks with the blood of Palestinians, with their fate, and dreams and aspirations for an independent Palestinian state," Abbas said.
Ahmed Yousef, adviser to Ismail Haniya, head of the Hamas government in Gaza, told Al Jazeera that Fatah is the real obstacle to Palestinian unity.
He said Hamas has urged Abbas repeatedly to reform and restructure the PLO to make it more inclusive, but without success.
The remarks from both sides come just days before Hamas is due to hold talks with Egyptian authorities on extending the ceasefire between the group and Israel.
Meshaal in Tehran
Meanwhile, a delegation from Hamas, headed by Meshaal, arrived in Tehran as part of a regional push to reinforce support for the group.
Iran's Supreme Leader Ayatollah Ali Khamenei told the visitors to be prepared for possible fresh Israeli attacks on the Gaza Strip, the semi-official Fars news agency reported on Sunday.
Meshaal, for his part, praised Iran for backing Hamas, Fars said.
"The Islamic Republic of Iran has a big share in our victory in the Gaza Strip," he was quoted as saying.
Meshaal has also met Mahmoud Ahmadinejad, the Iranian president.