Are food calorie values really integers?
They're not exact numbers. These numbers aren't exact for three reasons:

Each type of carb, protein, and fat has a different caloric value.  These are overall averages for each class.

Even if you were dealing with a single pure compound, the value couldn't be exact because there is individual variation in how much of that compound is metabolized based on digestion, absorption, etc.

Even if #1 and #2 were not issues, the number still couldn't be exact, because it is a measured value, and measured values are never exact.  Only counted numbers, and values that are exact by defintion (e.g., the speed of light) are exact.


To get a better idea of how these values are determined, and the issues and uncertainties associated with measuring them, I'd recommend reading through:
Food and Agriculture Organization of the United Nations. FAO food and nutrition paper. Chapter 3:  Calculation of the energy content of foods - energy conversion factors. Food and Agriculture Organization of the United Nations, 1979.
Here is a direct link to chapter 3:
http://www.fao.org/3/Y5022E/y5022e04.htm