Hamas and Fatah reached an agreement earlier this month in the Saudi capital Mecca after months of political deadlock.
The crisis spilled over into open conflict in which more than 100 people were killed.
Abbas has asked the current government to continue until a unity administration can be formed.
|Armed groups have clashed |
on Gaza's streets [AFP]
The Palestinian president said: "I call upon you to respect Arab and international resolutions, and the agreements signed by the Palestine Liberation Organisation."
Hamas has never formally joined the PLO.
Haniya said he would "work in accordance" with Abbas's letter of designation.
It was unclear whether this went far enough towards meeting the demands of the quartet of Middle East negotiators (the US, EU, UN and Russia) that Hamas recognise Israel, accept past peace deals and renounce violence.
However, the US has said it will continue to boycott all the new government's ministers, including non-Hamas members, unless international demands on policy towards Israel are met, a Palestinian official and diplomats said.
Tony Snow, a White House spokesman, said: "Our conditions always remain the same."
"Those are the basic conditions and we certainly hope that Israel will find that partner we've worked with president Abbas and we continue to do what we can," he said.
The move would be a blow to Palestinian leaders, who hoped a unity government would lead to an end of a crippling Western aid freeze.
Condoleezza Rice, the US secretary of state, is to travel to the Middle East to hold joint talks with Abbas and Ehud Olmert, the Israeli prime minister, on Monday.