On one website I saw a sentence "My friend calls me every day. I'd like it if she stopped." Is it correct to say "I'd like her to stop it" instead of "I'd like it if she stopped"? Which is more natural and correct?
Although synonymous in meaning, they're two different phrases created in slightly different ways.
"I'd like her to stop it". This is saying that you would like an event to occur. Other examples include, "I'd like a sunny day tomorrow" or "I'd like a super double pepperoni pizza please".
"I'd like it if she stopped". This says something else: you would like 'something' if an event occurred. The object is this abstract 'it' and the event is her stopping. Other examples: "I'd like Mel Gibson if he hadn't made so many bad films" or "We wouldn't like a pepperoni pizza if none of us would have it".
To answer the question as to which is better, I'll need to explain what I mean by "abstract 'it'". The 'it' is talking about the person's current situation. Example: "I'd like my current situation if she stopped", or "I'd like the state of affairs if she stopped".
As for which to use... this is more a matter of preference. You may notice that the two different constructs show emphasis on different things:
"I'd like her to stop it" puts more emphasis on the need that she stops what she's doing.
"I'd like it if she stops" puts more emphasis on the fact that like it if she stopped it.
That may have made it much more complicated than it really was, but, as is normally the case when one endeavours to work out how they've constructed their words, it was a journey of realisation for the author as well.