Saudi Gulf affairs minister Thamer al-Sabhan, in an interview with Al-Arabiya TV, said the Lebanese government would "be dealt with as a government declaring war on Saudi Arabia".
Sabhan said this message had been delivered to the Saudi-allied Lebanese politician Saad al-Hariri, who declared his resignation as the country's prime minister on Saturday in a broadcast from Saudi Arabia.
Hezbollah acts of "aggression" on the Kingdom "were considered acts of a declaration of war against Saudi Arabia by Lebanon and by the Lebanese Party of the Devil", he added.
Hariri cited an assassination plot against him in the statement announcing his resignation, and launched a scathing attack against Iran and Hezbollah for sowing strife in the Arab world.
The crisis has pitched Lebanon back into the forefront of a regional struggle between Saudi Arabia and Iran, which has also played out in Syria, Iraq, Bahrain and Yemen.
Earlier on Monday, President Michel Aoun, a political ally of Hezbollah, appealed for national unity following Hariri's resignation which toppled a coalition government that included Hezbollah and plunged Lebanon into political crisis.
Parliament Speaker Nabih Berri, another political ally of Hezbollah, said in a televised statement after meeting Aoun it was too early to talk about forming a new government.
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