Which is correct: 'one and half gallon' or 'one and a half gallons'?

In English plural is more than one, so gallons. In some languages (German IIRC) plural is two or more.

It is plural simply because it is more than one

the basic rule is that the singular is used solely for a singular item, for ONE item. Everything else gets the s at the end. In other words, the real rule is everything is plural except when singular, rather than singular gets changed to plural when there is more than one (which is how most people think of "plural" versus singular).

One of the few situations where you could say one and a half gallon would be as an adjective phrase or noun phrase, as in a one and a half gallon container, or a half-plate of food. both of those are singular because despite being a non-unit amount, they are a single group unit. a half-plate is not a half of a plate, it is a plate filled halfway, a half-plate; you can order seven half-plates or one half-plate (assuming the diner offers half-plates).

one and a half (1-1/2) gallons (plural) is correct.

One and a half GALLONS.

If it was one gallon ONLY then that is singular, so it would be GALLON. Anything more than ONE is plural.

it is how you use it in a sentence
or you could say 12 pints

one and half gallon is correct.

One and a half gallon

nonplural as it is a half gallon