Intensity is the physical correlate of loudness, and is also a correlate of stress in some languages. Moreover, stress can create differences in meaning in some languages (e.g. PRO-test vs. pro-TEST); we say that such languages have lexical stress, or that stress is lexically distinctive. It follows that if we can find a language where intensity is a good correlate of stress, and stress is lexically distinctive, that would answer your question.
One example is Papiamentu (Remijsen and van Heuven, 2005), and I'm sure there are others.
Note: I'm responding to the question in your title. The question in your main text adds the qualifier only, which makes such cases a bit harder to find, because intensity usually isn't the only cue for stress. Although I doubt pitch is the only cue for tone in your Mandarin example either, since tone also affects duration...
Remijsen, Bert & Vincent J. Van Heuven. 2005. Stress, tone and discourse prominence in the Curaçao dialect of Papiamentu. Phonology 22(2). 205–235.