Connecting Arduino to internet through HC-06 and then smartphone
Yes, but not in quite the way you think.

In this case the smartphone is acting exactly like the PC when you have the Arduino plugged in to it. You have a serial connection. Theoretically you can:


Send a request to the smartphone over the serial bluetooth connection
Write some software on the smartphone to interpret that request and perform some action depending on what the request is


That way the smartphone is then acting like an incredibly expensive and bulky ESP8266.