I'm sure sites won't close, and will find alternatives to cookies.
However if the solution to cookies is "in-browser", it's useless to move overseas.
If the purpose of a cookie is just to pass a variable to another page, it's easily replaceable by passing variables as url arguments, e.g. ?x=1&y=2 (funny thing is there's a microstock site which even passes passwords this way in plain text, which is not very sensible)
And for tracking us, my guess is that sites will push even more on people voluntarily - or unknowingly involuntarily - giving away information (e.g. by clicking on Like, Twit, Plus and similar or by installing apps).
I just hope they won't turn to compulsory registration/login as a way to track us, which is more annoying than cookies.