The Persian Gulf Countries only produce a percentage of the world's oil. All of OPEC together produces about 40~45%.
Being cheaper to produce (having greater profit at the same price) does nothing to prevent others from producing and selling oil. It does allow the price to fall further if the supply/demand curves shift that way. But as long as ConocoPhillips or BP can make money selling American Oil, it doesn't matter that Aramco can make more. They don't have enough production capability to shut out everyone else.
Oil is sold at market price. As long as the price stays above the cost of operations/exploration plus a profit, oil will continue to be produced in places like the Gulf of Mexico.
Opening up the rest of our waters would not take us to zero imports, but it would reduce the amount of dollars we send to places like OPEC.
The price of oil is low. The cost to extract oil miles down in the Gulf of Mexico is expensive and much higher than pulling it out of the mideast sand. Therefore those extracting oil out of the Gulf are at a competitive disadvantage than those pulling it out of the sand. So just allowing more drilling by itself is not a magic wand to making us independent of foreign oil unless the exploration and production cost differentials are dealt with. We are dependent on foreign oil not just because of environmentalists.