Is this correct in English?

Yes it is you go gurl

liable = likely to

He's a leader liable TO DO WHAT?

If you say a person is liable to do something, it means that he is likely to do it. If you say a person is liable for something, it means that he has to pay for any damage it caused. You never say that someone is a liable person.

There are mistakes throughout your question. 

First, "liable" is not in any way related to "responsible" or "competent".  Perhaps you meant "REliable"?

Second, "counties" seems to be a mistake too.  Did you mean "countRies"?  A county and a country are two different things.  I've never heard of a county being a member of a security council.  But countries are.

"resolution" is the best choice of the three, but I wold say "ABIDE BY the resolution...".  I suppose "decision" could be acceptable, but I don't think it's the best choice.

And third, you made another mistake in that sentence too.  It's "...of THE security council".

>> I wrote: He’s a liable leader. <<

Not really sensible. I can't imagine a context in which "liable" makes sense in that sentence.

>> All counties should adhere to the .......... of security council. <<

NONE of those are correct.

The sentence should read

- All countries should adhere to the [...] of the Security Council. <<

In THAT case "all of the above" would be sensible and grammatically-correct word choices.

No, it's not. That's not something people say, but if you were to say it, English speakers' inference would be that you're saying that as a leader, he's a liability, meaning he's a leader whose presence or behavior is likely to cause embarrassment, put those led at a disadvantage, and/or cause damage that others will then become liable for, like those he leads, stockholder and stakeholders of the organization in which he's serving as a leader, etc.