Database Mail sending functionality not working on local system
I'm not sure what you are talking about with "gmail profile". SQL Server Database Mail has its own profiles and has nothing to do with any email clients on your machine.

When you create a Database Mail profile (SQL Server Management Studio, Management, Database Mail) you must specify the ip address of the SMTP server or servers to use. What you are confiruing is the "outgoing email server" for Database Mail (similar to configuring an Internet email client like Thunderbird or Windows Mail). Those SMTP servers must be configured to allow connection and to allow relaying from the SQL Server machine's ip address, something the email administrators will be able to do. 

That error message suggests to me that either the above has not been configured or that there is a firewall between the SQL Server machine and the SMTP server that is blocking the traffic.