The difference between 'watch' and 'see' is this:
* 'See' is a sense, and is involuntary. If you are not blind, and your eyes are open, you can see.
* 'Watch' is a conscious activity. You choose to watch something, and you pay attention to what you are watching.
(NB The difference between 'hear' and 'listen' is exactly the same - 'hear' is a sense, while 'listen' is a conscious activity involving a degree of attention.)
If you are talking about a hobby or general preference, you would say:
"I like to watch movies."
In some cases, it is possible to use 'see' with movies - as in 'I've never seen that movie', or 'I saw a good movie last week'. However, if you are talking about something which you enjoy doing, this means that you are focusing on the enjoyable activity itself, and the activity is 'watch' , not 'see'.
By the way, it is important to use the PLURAL form of the object, not the singular. 'I like to watch movie' is incorrect (or 'Chinglish').
I like to watch movies.
I like to eat bananas.
I like to wear hats.
Remember to use the plural form of countable nouns in sentences such as this.
I hope that helps.