A relaxation oscillator requires a discharge device or circuit which exhibits hysteresis: this means that when the capacitor charges to the high threshold, the device turns on a discharge path and keeps the discharge path open even as the capacitor drops below the threshold until some low threshold is reached at which point the discharge path closes.
The discharge transistor you have there does not have hysteresis.
Furthermore, the circuit has stabilizing negative feedback. The discharge transistor acts as an inverting stage (which is why it is negative feedback even though it goes through the + terminal of the op-amp). Turning on the discharge transistor more causes it to conduct more collector current, draining the capacitor to drain to a lower charge, which has the effect of opposing the turn-on of the transistor.
This arrangement will naturally solve for a capacitor voltage which is precisely so high that, through the op-amp, via a minute voltage difference between + and - inputs, it programs the discharge capacitor to have a collector current which matches that of the charging capacitor. The charge current continues to flow, and just enough of it trickles into the capacitor to keep it charged in the face of any leakage. The rest goes through the discharge transistor's collector-emitter path. Any build up of extra charge on the capacitor is amplified through the comparator's monstrous gain, and almost instantly raises the collector current on the discharge transistor, which will promptly remove the charge. If the capacitor loses charge, the discharge transistor will instantly reduce its collector current, causing more of the charging current to be diverted into the capacitor to replenish the charge.
Such system as described could oscillate, but not as a relaxation oscillator. There would have to be some frequency at which the product of the gain and feedback is 1, and the feedback is phase shifted by 180 degrees. That would be a phase shift oscillator.
Without hysteresis, and without the conditions for phase shift oscillation, your circuit is simply stable. It settles back to equilibrium even if externally disturbed.