Receiving SMS message to GSM sheild without Serial Monitor?

I'm putting together a project in which I hope to be able to send a SMS message to an arduino and that message will be transformed into Morse Code. ( LED blinking or vibrating motor, etc. )

I won't be able to have the arduino and computer connected through the USB port and therefore will not have the serial monitor open.

THE QUESTION: Is is possible for the program to run and for SMS messages to be received ( and transformed into Morse Code ) if the arduino is not connected to the computer and Serial Monitor?

Appreciated the feedback. Just want to figure that out before buying a GSM shield and trying to work out the code for all that.

You connect arduino and gsm.
Arduino send commands to gsm and can read the answer on it.
Arduino can monitor inbox from gsm.
You do not need external console output to control it.

You can make a fully autonomous unit without external PC and other devices. Just a bunch of Arduino+GSM. Send sms-message from phone to this unit and then blank with led on it. No computer needs for control this (only for debuggin may be).

But GSM may need external power 5V 2A - consider it if you plan to be powered by batteries.

Thanks for that and the reminder about the 2A supply.

If I do eventually make the project I'll post the results.