Can an LC oscillator be used to generate visible light?

Small-value inductors and capacitors are possible, but you also need extremely small size else the lumped system approximation your equation relies on becomes invalid. Let's say the wavelength of the light to produce is 700 nm. A distance of half that (350 nm) will cause an inversion (180° phase change). Generally, you want to keep the maximum dimension to $\frac{1}{10}$ the wavelength to consider it a lumped system without having to worry about that further.

Your inductor and capacitor, therefore, have to all fit roughly within a 70 nm dimension. With everything that close to everything else, there will be a lot of parasitic capacitance. Getting a low enough capacitance and still have the deliberate capacitance across the ends of the inductor dominate over all the distributed stray capacitance won't be easy. Even if you could do that, you still have to drive it with a 430 THz signal somehow. Where are you going to get that from?

If, by LC oscillator, you mean a circuit composed of an 'ordinary' inductor and capacitor etc. then the answer is no.

The equations for the LC oscillator are derived within the context of ideal circuit theory which is the limit of a number of assumptions.

To apply the results from ideal circuit theory to physical systems, the physical systems must approximate the assumptions of ideal circuit theory.

One of these is that physical circuit elements can be represented with lumped element models.

And, for that to hold, the wavelengths of the signals of interest must be much larger than the dimensions of the physical circuit elements.

When this assumption does not hold, we must use the distributed element model such as in, for example, transmission line theory.

Another related consideration is that physical circuit elements have parasitic properties that cannot be avoided. Thus any physical system of conductors possess parasitic inductance and capacitance that become significant at high enough frequencies.

For visible light, the wavelengths are so small (hundreds of nanometers) that the assumptions of ideal circuit theory are not remotely valid so, unless one generalizes the notions of "inductor" and "capacitor" (and the other circuit elements) far outside the ordinary, the answer is no, an LC oscillator cannot produce visible light.

During WWII, British radar researchers struggled to develop very short wavelength radar waves (on the order of 10cm). These would be useful because they would (among other things) permit small airborne antennas for night fighters, bomber defense and navigation, etc. They couldn't do it with ordinary electronic circuitry, and had to develop the cavity magnetron to generate such high frequencies and high power levels necessary, which uses much different principles. Now take that difficulty, and try to generate Infrared waves, much less even shorter Visible light. That's not even considering the problems of building an antenna that tiny!