Yes, you can and it is perfectly legal. You can have sex in your bed, in your bathtub, on the sofa, on the rug in front of the fireplace, in your Jacuzzi and in your car parked in your garage. The problem your friend had was having sex in public places. You wrote:
The cop would not have intruded into your friend's private garage without a search warrant, would he? He was patrolling a public place.
I don't want to be moralistic, but it is good to be accurate.
But getting to the point, if you are going to move to a new country, and I did, then you have to accept that country's basic laws. Once you become a citizen of your adopted land, you should have the right to mold that country to you beliefs, by legally accepted methods but not by violence.
If your chosen adopted country is the USA (not my adopted country) then, IMO, asking for the abolition of the Constitution in favor of Sharia law is treason. What should be legal would be to seek constitutional amendments to incorporate your beliefs into the Constitution via amendment. The safeguards built into American law would make it very hard to get those amendments ASAP. If you think that Americans want instant gratification, you have seen nothing, Radical Islam wants instant gratification even worse.
Three Cheers for Australia doing the right thing to defend itself.