But the governing coalition, which made the decision on Saturday, said it will press on with the elections even if Lahoud does not give the presidential approval required by the constitution.
"Our decision is to hold the elections," Aridi said.
Lahoud has 15 days to approve the cabinet decree calling for by-elections. Political sources say a move by the cabinet to hold the elections without Lahoud's approval might encourage him to appoint a new cabinet, leaving Lebanon with two governments.
The assassination of Eido, killed with nine other people on Wednesday, reduced the majority of the anti-Damascus coalition to four.
The governing coalition, which has strong US backing, won its majority in elections following the 2005 withdrawal of Syrian forces from Lebanon.
Syria was forced to pull out of Lebanon due to international pressure and mass protests after the February 14, 2005 assassination of Rafik al-Hariri, the former prime minister, whose killing many blame on Damascus.
Syria denies involvement in the killing of al-Hariri and other anti-Syrian figures including Eido, a Sunni Muslim politician in Beirut, and Gemayel, who was a Maronite Christian legislator and a cabinet minister.
Lebanon's political system guarantees representation for the country's array of religious communities by allocating government posts and parliamentary seats on the basis of sect.
Lahoud and the opposition have said the cabinet lost its legitimacy in November when ministers from Hezbollah and Amal quit, stripping the government of all its Shia Muslim representation.
Hezbollah and Amal, which represent the vast majority of Lebanon's Shias, are allies of Syria.