Is there any way to predict/calculate the pH level of a recipe when the level of individual ingredients is known?

Short answer — it's definitely not that simple. For one thing, the pH scale is logarithmic, not linear. For another, almost every acid and base you're likely to encounter in a kitchen is “weak” - meaning there's an equilibrium between neutral and ionized forms. When diluted, the equilibria shift. When the temperature changes, the equilibria shift. When mixed, the equilibria interact with each other.

If you're working for a big industry and filling railroad cars with your recipes, and doing the same reactions again and again, and trying to maximize profits, it could be worth doing the math. But for a small scale cook - you could probably get as far as accurately predicting the pH of a mixed drink, but not a lot further, I think.

There are a number of problems with this:

First, pH is a log scale, as RalphMudhouse explained. This means you cannot simply average the pH of your components--it's a lot more complicated than that. Still, this point would suggest that you could somehow calculate the pH, except...

pH is only well-defined for water-based solutions. The definition of pH is based on the concentration of hydronium ions in a water based solution (other solvents not really being relevant for this particular question). Since food is usually a mixture of states (stews, breads, etc.), the concept is ill-defined: the different components of a stew might have different pH, for example.

Additionally, stuff reacts. chemically, food is very complicated, and reactions will happen that may affect the pH. Different ingredients may behave differently, and the resulting reactions are unpredictable for a quantitative, a priori calculation.

However, pH is still important. Just because it can't be calculated doesn't mean it's not relevant: it will still affect stuff like browning. In my experience, you can still gauge acidity by simply tasting as you go: the more acidic something is, the more sour it is (basicity is a bit harder, and I don't have a good solution for that). Also, I don't think a calculation is actually that necessary--a simple intuition of things being basic or acidic is more than enough for home (or even restaurant) cooking.

You can purchase professional pH meters online from modernist cooking websites, eg. Modernist Pantry. They are somewhat expensive, however. I don't have one myself, but I might invest in one if I ever get into canning / spherification / other pH sensitive techniques.

Note that if your acids and bases react, there's no guarantee that your overall pH would combine linearly; that's why measuring tools exist.