Yes, you could. Rocket scientists tend to be stuck on SI, though, and use joules. (Also, I'm doing an exchange year in Germany at the moment, and all the food in my cupboard has its energy content listed in kilojoules, not Calories).
Also, you have to be careful with the spelling of the words calorie and Calorie.
"calorie", with a small c, is as you have said, the amount of energy required to raise the temperature of one gram of water by one degree (the gram was originally defined as the mass of one milliliter of water at 0 C).
"Calorie" with a big C is the unit used for measuring the energy content of food, and is equal to the amount of energy required to raise the temperature of one *kilogram* of water by one degree, or 1000 small-c calories.
It's confusing, I know.
But you can measure the energy content of anything in any unit of energy you wish: The Calorie, the calorie, the joule, the gram*c^2, the electron volt, the erg, the firkin*(furlong/fortnight)^2, the horsepower-nanoparsec/c, the gigaparsec*atomic-mass-unit-force, or the megaton TNT equivalent.