Checks and Balances was designed to keep each branch from assuming too much power. Presidential power is checked and balanced in many ways. While he can veto various bills and keep them from becoming laws, he can be overridden by a 2/3 vote in both houses. SO if the Congress feels like the President is over stepping his bounds then they can in turn overrule him. The Supreme Court is also in the mix in that if they deem a law to be unconstitutional then they can rule it so and the law is no more. Presidents can influence Supreme Courts by whom they nominate for open positions (or in the case of one President, attempt to add more judges to the Court). However, after the nomination is made and the Senate approves and confirms them then they are free. The only way a Judge can be removed is they are found to be doing something illegal, deemed incompetent and unable to continue doing their job, resign, or die. This allows judges to "operate outside the realm of politics." The idea is that they can focus and not worry about politics and we can assume that decisions will be made free from political influences and bribery.
The main difference between ancient Greek cities and modern US states is that Greek city states were ruled by kings obviously because they didn't yet have the true concept of democracy - D. They also had their own laws but so do US states.
