What is the relationship between a digital audio signal and sound pressure level?
As you say, pressure will be proportional to output level, if the output level is equivalent to voltage. But sound pressure level (SPL) is by convention expressed in the logarithmic decibel (dB) scale, with 0 dB referenced to some standard sound pressure level. The relationship is:

$$\text{SPL} = 20\log_{10}\left(\frac{A}{A_\text{ref}}\right) \text{ dB,}$$

with a reference output level $A_\text{ref}$ that would give an SPL reading of 0 dB. For measures of power such as $A^2$ the formula changes slightly:

$$\text{SPL} = 10\log_{10}\left(\frac{A^2}{A^2_\text{ref}}\right) \text{ dB.}$$

With $A$ and $A_\text{ref}$ non-negative, the equations are equivalent with the square inside the logarithm or outside it as a factor $2$.