Grammatically, "It is fun to write English letters." is correct. But is the following also grammatically correct?
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.
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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.