Hi,
simple answer is no. The islamic law is based on the texts of the Quran and Hadith - Sunnah the life of teh Prophet. the laws derived from this are the shariah. Obviously to live according to the Quran you require muslim believers to implement the shariah, at macro and micro govermental level.
Although there are some ideas that the us constitution and laws share with islamic morality and principles . It cannot be described as a nation living according to the Quran. Furthermore i feel there is no nation in the world today living according to the Quran to a significant extent, from top to bottom.
Furthermore, one could talk about a simillarity on a very general level : the "basic rights of humanity" as the west describes them, which are something explicitly mentioned in the Quran and Sunnah in much detail and are very important to muslim life in this world and the next.
I will post a thread on this later.
Allah knows best.