I can imagine someone saying, "I don't have a problem with spicy, I just don't prefer it." I think your understanding in this context is good--that he wouldn't get sick or be bothered by it. He can handle spicy food.
To say, "I have no problem with English," can mean two things.
1) That it doesn't bother you. This is not what you mean to say, but is how I would understand it from a native speaker.
2) That you can speak and understand English very well. This would be better stated as, "I don't have any problems speaking English." Clarify exactly what you don't have a problem doing.