In English, American English tend to use quotation mark ("he"), and British English tend to use single mark ('he'), is it correct?
American English: He said, " I'm eating."
British English: He said, ' I'm eating."

Question

In English, American English tend to use quotation mark ("he"), and British English tend to use single mark ('he'), is it correct? 
American English: He said, " I'm eating."
British English: He said, ' I'm eating."

Answer

I think traditionally, British English has used single quotation marks, and double quotation marks when quotes are nested. I believe this is still the case in many but not all, maybe not even most, publications.

I checked the websites of 
The Guardian newspaper, The Times of London, The Independent, and The Daily Mail. I noticed that all except the Daily Mail are using double quotes in the body of their articles these days. The Irish times does this as well. So it looks like English is becoming more consistent around the world. (Traditionally, UK, Ireland, and South Africa followed the single-quote tradition.)

Canadian, American, Australian, and New Zealand publications all use double-quotes (except single-quotes when nested).

Interestingly, I never noticed this before but in all countries, newspaper headlines use single quotation marks even if the body of the article uses double quotation marks.

So I agree with John's answer. You're probably better off using double-quotes in general unless your teacher or editor prefers single-quotes.

Answer

You are kind of right.

In both dialects, they use both " and ', but British is more lenient.

In American English they use double quotations, but if there is a quotation within a quotation, they use the single version. For example: "He asked me what 'The Big Apple' was".

In British English, they can use single quotations without the double ones. It all depends on their editorial style.

But generally, when reporting a speech, it's best to use the double quotation marks. ("")