Tonality is a system of composition (or the theory describing it), and one aspect of Tonal music (that is, music following Tonality) is that there is a pitch center. However, Non-Tonal music, that is music composed outside the system of Tonality, can still have a pitch center, though it needn't. Atonal music is composed to avoid the sense of pitch center.
The answer to your question is that Tonality, by its rules of construction, depends on consonance. However, you can create a pitch-centric piece comprising only dissonances. For example consider the following composition:
Each vertical pair is played simultaneously. If you emphasize the lower notes, C will sound like the pitch center; If you emphasize the upper notes, D will sound like the pitch center. Every dyad is dissonant, yet the ear perceives a pitch center because the "horizontal" pitches follow a directed pattern. So this piece can be heard as pitch centric, but non-tonal.
Now, just to muddy the waters, my saying that the above dyads are "dissonant" somewhat abuses the term. My intervals are dissonant as dissonance is defined in the Tonal system. That is, in another music system, or to someone unfamiliar with tonality, my composition might be quite pleasing (doubtful, but ...). Further, my composition achieves it's sense of pitch center by relying on the (horizontal) use of familiar Tonal patterns (i.e., short ascending and descending scales). Had I ended the composition one dyad earlier (on E/D) a listener familiar with Tonality would likely hear it as ambiguous, whereas another listener might hear the emphasized final pitch as a perfectly satisfactory "center".
Dissonance, ultimately, is in the ear of the beholder.