I want to transfer 4GB .tar.gz file from Hostgator to DO, and I am unable to access that file via URL (so I can't use wget), also its not possible to do via FTP. So how can copy that file to DO?
Since hostgator provides ssh access you can use the scp command to transfer the file directly from your hostgator account to your new droplet.
scp user@hostgatorserver:/path/to/your/file.tar.gz root@dodropetip:/path/on/droplet/.
Replacing the ssh details with those of your hostgator account and new DO droplet and using this command will allow you to transfer the file without making it web accessible.
the best way is to make the 4GB .tar.gz available on your apache2 and then do a php script that download it :)