There's no way they can effectively close it. Any weapons and personnel brought forth would be eliminated in short order. Their 'navy' couldn't defeat the local yatch club.
All it would take is to sink one supertanker there. The threat of another would effectively stop all traffic......
Note the shallowness of the waters. It's not the width of the strait, it's the depth of the water that's the threat. Iran could sink a tanker there before anybody could do anything about it. And even if we retaliate, the damage is done........