The US (and others) have minesweepers and I would bet they are there or heading that way.Based on the analysis I saw earlier today the most effective means they have to close shipping would be to mine the shipping lanes which would be an act of war in my opinion. Trump has opened up Pandora's box now and should be committed to deal with the threats Iran presents to the global economy.
Clearing mines is not instantaneous, but is possible with time.
Until mines are cleared, however, supertankers cannot sail through the strait, not because it is impossible, but because maritime insurance cannot be secured until the mines have been reliably cleared.
You are correct, laying mines is an act of war, but so is dropping MOPs on a country. This would be a counter-escalation to the American escalation. I think the American counter-counter-escalation would be removing the Iranian minelaying assets along the coast.