Can a President dismiss a member of the US Supreme Court and replace him with someone more in agreement with the President?

No, that would be unconstitutional. Article III of the Constitution provides that Supreme Court justices are appointed for life. One reason for this is to prevent each new Presidential administration from sweeping the Court of opposition. Doing so would eliminate the independence of the judicial branch, simply leaving it a political extension of the Executive branch, something the Framers believed would result in tyranny.

Supreme Court justices can only leave the Court in one of four ways:

1. Resignation
2. Retirement
3. Death
4. Impeachment (and conviction)

The House of Representatives can only initiate impeachment proceedings for legitimate reasons, as outlined in the Constitution. Justices cannot be removed from the bench simply because other members of government disagree with their decisions.