Thousands of people have already fled the southern city after the Israeli military began what it called a "limited" operation on Monday.
US President Joe Biden has repeatedly warned against the operation, saying that it would cross a "red line".
But Mr Netanyahu dismissed the US warning, saying Israel would fight on.
"If we need to ... we will stand alone. I have said that if necessary we will fight with our fingernails," the prime minister said.Mr Netanyahu also invoked the war of 1948 - where the newly formed State of Israel was attacked by a coalition of Arab states - to dismiss the US warnings.
"In the War of Independence 76 years ago, we were the few against the many," he said. "We did not have weapons. There was an arms embargo on Israel, but with great strength of spirit, heroism and unity among us - we were victorious."
Despite Mr Netanyahu's comments, his government has come under pressure after the US suspended the delivery of 1,800 2,000lb (907kg) bombs and 1,700 500lb bombs over fears that they could cause civilian deaths in Rafah.
Mr Biden went further still in an interview with CNN on Wednesday, saying that if the attack went ahead he would further suspend supplies of artillery shells and other weapons.
Yoav Gallant, Mr Netanyahu's defence minister, dismissed the US warnings, saying Israel's "enemies as well as ... best of friends" should understand that his country "cannot be subdued"."We will stand strong, we will achieve our goals," he added.
The comments came hours after the UN said more than 80,000 people had fled Rafah since Monday amid constant bombardment and as Israeli tanks massed close to built-up areas.
The UN also warned that food and fuel were running out for the more than one million still sheltering in the city, because it was not receiving aid through nearby crossings.
Israeli troops took control and closed the Rafah crossing with Egypt at the start of their operation, while the UN said it was too dangerous for its staff and lorries to reach the reopened Kerem Shalom crossing with Israel.
Israeli forces said they were conducting "targeted raids" against Hamas elements remaining in the city. However, the Israeli government has refused to rule out a full scale invasion, leading to Mr Biden's warning he would not supply it with the munitions to do so.White House national security spokesman John Kirby said Mr Biden did not believe "smashing into Rafah" would advance Israel's objective of defeating Hamas.
"An enduring defeat of Hamas certainly remains the Israeli goal, and we share that goal with them," Mr Kirby said.
"The argument that somehow we're walking away from Israel, or we're not willing to help them defeat Hamas just doesn't comport with the facts," he said.
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