If I get a positive response on a Covid-19 test for the purpose of travelling to the USA, and then do another and get a negative, can I use that one?

I’m planning to travel to the USA and I feel concerned about the false positive rate of antigen tests for Covid-19.

You should feel concerned about the false negative rate, not the false positive rate. It is shocking that any country allows intercontinental travel without a negative PCR test.

False positive — worst that can happen, your travel doesn’t occur. Nobody gets hurt.

False negative — worst that can happen, you bring a new variant to another continent and that variant kills a million people. More likely, you infect some fellow passengers or people in the destination country with an already-present variant, and you hospitalise just a handful of people, some of whom might recover without long covid.

Can you travel after a positive test? Maybe. You can do many things that are either illegal or immoral or both.

Should you travel after a positive test? No, most certainly not. Even if your positive test is followed by two negative tests, JohnatanReez’s answer illustrates there’s still a 1.3% chance you are actually positive, which makes travelling entirely irresponsible. If 400 passengers reason that way, that would mean on average 5-6 infectious passengers boarding a plane. How many are infected after a 10 hour flight is anybody’s guess. FFP2 masks are great, if worn continuously and perfectly by all; that condition may not be met on an intercontinental flight, and 10 hours is longer than what those masks are designed for — so it would be wrong to think your infection state is not important as long as you’re wearing a mask.

This may depend on the country you are coming from, but in general no. I’ve written the answer with the example of the UK, but similar rules apply elsewhere.

If you test positive on a pre-departure test, you must follow the required next steps for someone who tests positive in the country you are in. In the UK that would mean self-isolating and getting a PCR test for confirmation. If you test negative on the PCR test you can stop self isolating. I am not aware of any country where the appropriate next steps are to take further lateral flow tests until you get lucky.

In many countries, Covid-19 is a notifiable disease, meaning that the organisation which tests you is obliged by law to report positive tests to the local health authorities.

Short answer, no, as per Undo’s answer above & CDC regulations. It would also likely be considered misleading to do so, which is expressly prohibited.

Any passenger who fails to comply with the requirements of section
2, “Requirements for Aircraft Passengers,” may be subject to
criminal penalties under, inter alia, 42 U.S.C. 271 and 42 CFR
71.2, in conjunction with 18 U.S.C. 3559 and 3571. Willfully giving
false or misleading information to the government may result in
criminal penalties under, inter alia, 18 U.S.C. 1001.

Some additional notes regarding the practical validity of a positive vs a negative result:
Before you start showing symptoms, false negatives are 100% when testing for the virus. Viral load (a strong determinant of transmission risk) reaches its maximal level two days prior to symptom onset. Additionally, 50% of cases are asymptomatic.
Even more broadly, a study on false negative rates found:

• Negative rates were 100% two days prior to symptom onset
• Negative rates were a minimum of 25% two days after symptom onset (or date of maximal viral load)

It is likely your negative result has a minimum 25% chance of being a false result based on this research.

Sources:

Temporal dynamics of viral load and false negative rate influence the levels of testing necessary to combat COVID-19 spread

Recalibrating SARS-CoV-2 Antigen Rapid Lateral Flow Test Relative Sensitivity from Validation Studies to Absolute Sensitivity for Indicating Individuals Shedding Transmissible Virus

CDC Global Testing Order

You didn’t ask whether you should do this, you asked whether you could. This is a question answered by simply looking at the actual CDC order:

this Notice and Amended Order prohibits the boarding of any passenger [..] unless the passenger:

Presents paper or digital documentation of one of the following requirements:

(i) A negative pre-departure viral test result for SARS-CoV-2 conducted on a specimen collected no more than 1 calendar day before the flight’s departure from a foreign country (Qualifying Test)

The order goes on to discuss past positive tests as a way to show recovery, but there is no mention of a positive test superseding a negative test taken in the last 24 hours.

According to the order, you need only to produce a negative test taken in the previous 24 hours. If you can do that, you can board your flight.

Use your judgement, as always.

There’s a saying

A person with one watch always knows what time it is.

A person with two watches never knows.

The basis of this saying is that if you have two watches you don’t know which one is correct.

In your case you posit that if the first test shows a positive and the second test a negative, then you should go with the negative test. The trouble with this logic is that you don’t know which test gave a false result. Was the first test a false positive? or was the second test a false negative?

Ideally if you tested positive with one test, I’d want to see multiple, consistent negative tests before assuming that the first test was a false positive.

Pragmatically I would treat a positive test as meaning you are infected and that you should self isolate, no matter what a later test says. While it may screw up your travel plans, it does the most to protect your fellow travelers against potentially being infected by you.

As an example of your logic, (and not making a political judgement), prior to the (1st?) US presidential debate for the election, Trump apparently tested positive and then on a subsequent test, tested negative. The negative test was the assumed state, yet a short time later he was in hospital getting treated for Covid.