How would you defend ancient Rome against America’s modern conventional military?

Persuade America’s military that Carthage is the real problem.

I think I have seen that one.

Best case scenario is if you assume that the enemy force has no access to reinforcements or supply replenishments.
First thing you do is relocate the political leadership to concealed locations, as the Roman Senate will be the first target of enemy airstrikes.
Then you send in expendable waves of Roman Auxiliaries to force the enemy to deplete their ammunition reserves.
Then it should hopefully be a case of conventional ancient warfare. Modern body armour is pretty much useless against bladed weapons.

Offer free massages to soldiers who defect. Or play politics with the enemy’s political structure.

The enemy will expect to totally crush you. You need to do something unexpected.
Start digging tunnels and massive canals from the Mediterranean Sea. The goal is to flood them out with a deluge of water.
Or make chemical weapons – lead fumes are handy.
Poison the land they must cross by contaminating all food stuffs with vegetable poisons like belladonna from henbane, deadly nightshade, hemlock, and hellebore. If they resupply with your food they will be incapacitated.

Negotiate immediately for terms!

Trade gold to Russia in exchange for protection. Also the ten best sex slaves to Putin.

Pray to their gods for divine intervention.

Every Roman soldier must dress in civilian clothes. There should be no uniform or indication of rank.
Make many small skirmishes all over the countryside.so the large force must be constantly wary. They will be forced to send out smaller patrols to run down the skirmishers.
Attack patrols viciously to acquire some advanced weapons .
Raid supply lines. Use fire as a WMD whenever possible and to acquire weapons.
Slowly use those weapons to acquire more. Move them quickly to another area so there is an element of surprise.
Rome cannot win an all out attack. It needs to wear down the hearts and minds of the soldiers.

Genius @LuckyGuy Rome doesn’t have to win. They merely have to hold out until the war is no longer politically feasible in the United States—the enduring lesson of Vietnam.

Invoke international law, and the laws that restrict US military action. Deploy the military amongst civilians.

A wipe out. Modern technology opposed to First Century C.E. weapons? It would be like an alien invasion to us today. Romans were bad to the bone in their time and place, but what is a Centurion going to do against a drone strike, or modern tanks? Why not put the Wright Brothers against an F-16 jet fighter?

I would employ gorilla tactics.

Better hire King Kong!

How did the US military do against guys in sandals with IEDs and AKs in Afghanistan and Iraq?

Success means occupying and administering a territory, providing food and healthdcare.

The same way the Jews did with the Romans at Masada.

Wall ourselves in the temple?

I might use a different tactic. Let all Romans know the only way to defeat this enemy is to use guile and trickery. It is ery citizens job to work on this.
Appear to be welcoming. Encourage the women to fraternize with the soldiers. They are 19–21 year old males so that should be easy.
After months of apparently friendly relations the soldiers will be less cautious. On a certain pre-planned date, the Ides of March perhaps, all, and I mean all, citizens strike to kill and steal the weapons.

Come to think of it, there would be no cause to attack Rome, for without a doubt a few helicopters from the skies, the destruction of a single legion utilizing flamethrowers, grenades and machine guns. The ONLY possible conclusion for any witnessing such slaughter is that the gods have arrived and that resistance is both pointless and stupid.