Grammatically, "It is fun to write English letters." is correct. But is the following also grammatically correct?
It is fun writing English letters.
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Indef) To write English letters is fun. ~ It is fun to write English letters.
Indef('s)) Writing English letters is fun. ~ It is fun writing English letters.
This is not necessarily true with other predicates, which all have unique patterns, as the second link should make clear.
I suspect that what is behind the OP’s question is uncertainty over when to use to + the plain form of the verb and when to use the -ing form of the verb. In the examples given, there is little or no difference (although they are rather unlikely sentences in any formulation).
Elsewhere there are differences, as a good reference grammar designed for foreign learners will show. For example, in some cases the construction with to + the plain form of the verb, as explained in ‘An A-Z of English Grammar and Usage’ by Leech and others, describes a possible action, while the -ing form describes the actual performance of the action.