The only way you can do that is if you use a texting service (Gmail SMS maybe) that would allow you to do so, but as far as veiwing your VZW text messages online, no you can't.
If you log into your account online under the number you want to view, go to:
My Verizon>My Device>Send and Manager Messages
You can view the messages that are currently on your phone, however you cannot view the deleted ones.
To view the deleted ones you'll need to have had the Verizon Cloud back up on and you can find them in your cloud for up to 90 days.