As long as you are not spamming then you should be able to.

For gmail addresses you will need to add an SPF Record to your domain and preferably DKIM. To send mail you need an SMTP (Simple Mail Transfer Protocol) Server. Without this your server does not know how to send email.

I must stress that this should never be used to send unsolicited spam email. It annoys People. IPs end up blacklisted (and are difficult to get unblacklisted). It makes things difficult for legitimate users. and Depending on where you are and the location of the VPS it can be illegal.