Can the president fire a supreme court judge?

No, not directly. Supreme Court justices serve "during good behavior," which means they have a lifetime appointment unless they commit an impeachable offense.

Justices can be impeached by the House of Representatives and tried in the Senate if they commit "high crimes and misdemeanors," which include crimes, unethical behavior, actions damaging to the government, etc. If found guilty, the justice will be removed from office. They cannot be removed on any other grounds, including incompetency.