Judges last week issued warrants for the two men along with Hamas military commander Mohammed Deif, saying there were reasonable grounds to believe the three men bore criminal responsibility for alleged war crimes and crimes against humanity in Gaza.
Israel's government, along with Netanyahu and Gallant, strongly rejected the accusations.
On Wednesday, the prime minister's office said it had informed the ICC of its "intention to appeal to the court along with a demand to delay implementation of the arrest warrants".
It reiterated that Israel both denied the authority of the ICC and the legitimacy of the warrants against Netanyahu and Gallant.
Netanyahu's office also said US Senator Lindsey Graham had updated Netanyahu "on the efforts that he is advancing in the US Congress against the ICC and countries that have co-operated with it".
US President Joe Biden called the warrants "outrageous" last week.
"Whatever the ICC might imply, there is no equivalence - none - between Israel and Hamas. We will always stand with Israel against threats to its security," he said.
ICC member states - which do not include Israel or the US - are obliged to take action in detaining a wanted defendant if they are found to be in their jurisdiction.
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