Absolutely not. Even after the revolution the end of the monarchy was hardly inevitable. Hell, France might still be a Kingdom today if Henri, the Comte de Chambord hadn't rejected the throne over a flag. If anything, I'd argue that France being a republic today is more of a fluke, and that it ended up as one through a pretty random series of events.
The abolition of the monarchy really was the result of a series of poor/medicore Kings, the prevelance of reactionary revanchism and in my opinion, most importantly, natural disasters such as bad harvests (the revolutions of 1789, 1830 and 1848 were all accompanied by bad harvests, which was surely no coincidence).
Keepinig the monarchy around can be done with a POD as late as 1870 by having the Comte de Chambord accepting the Tricolor as the French flag, or as early as 1789 by having Louis XVI making better decisions during the Estates-General.
Although, I should add the caveat that, while the monarchy could survive, the Ancien Regime, in most circumstances, probably couldn't. By the time of Louis XVI the Ancien Regime had simply ceased working, as the economic crisis could simply not be solved without major, systemic change.
However there was no guarantee that the constitution of 1791 would fail, that the Bourbon Restoration would fail or that the July Monarchy would fail. Had some small things gone differently, things could have been drastically different.
If any of the later Bourbons had compromised and accepted a more British system of government, they could probably have held onto their throne.