Yes, the president can sign a bill that become a law.
A bill does become a law when a president signs it, but a bill can also become a law without the presidents signature.
A BILL does not automatically become a LAW. It must be passed and sent to the President. Only when the President signs the bill does it become law.See below link:
A law.
To become a law..
If a bill is vetoed it can become a law by being sent back to the House of Representatives and the Senate. If 2/3 of both the House of representatives and the Senate vote yes for the bill to become a law, it becomes a law without a signature.
The President's signature is needed for a bill to become law in the United States.
In the United States, when a bill that has been approved by both Houses of Congress, it is sent to the President. Thus a bill can become a law when the President signs it into law.
When it is signed by the president
the bill is made
IN the US, a bill that has been passed by Congress become law when the President signs it.
so that a crappy, pointless bill will not become a crappy, pointless law. ;)