Can the Senate overturn a US Supreme Court decision?

No. The Senate does not have the authority to overturn a Supreme Court decision because of constitutional separation of powers.

No single entity - not the President, Senate, House of Representatives, state Governors, nor anyone else - has the power to overturn a US Supreme Court ruling. Supreme Court decisions cannot be nullified by other parts of government.
  1. The Supreme court can overrule it's own rulings.

  2. The Constitution can be amended. This would require a two-thirds vote of both houses of Congress, and ratification by three-quarters of the states (actually, at least 39).

Optionally, Congress can rewrite a law to conform with Constitutional standards, which is the most frequent response to the Supreme Court overturning a law.

For more information, see Related Questions, below.