If I sent a username and the password to a website over
HTTPS, is it possible that someone on the net would save this request and re-send it later to authenticate on the website?
If someone just records encrypted HTTPS session and if the connection served over the secure TLS protocol version with the strong cipher suite, then the answer is no. Neither they will be able to decrypt your credentials nor replay traffic to create bogus session. TLS protocol provides message authentication and has its own replay attack protection.
Unless they hijack your session (at which point you wouldn't need the re-authenticate), no. When setting up a new session the TLS protocol will negotiate a new session key. So the intercepted authentication will be encrypted with a different key making it impossible for the server to understand what the attacker has just sent. For more info on how TLS works please read this question and answer.
is it possible that someone on the net save this request
Since HTTPS is usually end-to-end encryption that someone needs to be at one of the ends, i.e. in the client (bad browser extension or similar) or the server (compromised server) to get the unencrypted request. The encrypted data will be of no help.
In case of legal SSL interception in firewalls that someone can be also there to access the request in plain. Illegal SSL interception (man in the middle attack) is similar, but in this case you get a browser warning which you should not ignore.
re-send it later to authenticate on my web site
If the data are grabbed then they can be resent.