Short answer. Yes, they are related.
Long answer. These words both are descending from the same PIE root
reudh, which stood for indicating something related to red. I use the phrase something related to red, since no one knows for sure (and, alas, no one most probably will know in future) whether this root in its first meaning indicated colour and then some additional meanings (blood, red-coloured fruits or berries etc.) had been derived; or vice versa.
Actually, as far as I know, red is one of the few colours (if not the only one) for which we have solid evidence of common PIE origin. English words
rusty, Russian words
руда (in some dialects of Old Russian this also meant "blood" by the way),
ржавый, French word
rouge, all of them are related.
The longest answer I can provide. French is a Romance language but its phonetic evolution was very intensive. If one would be asked to characterise this evolution in a single word, it wouldn't be a too bold oversimplification to say reduction. For example, the French word août (which is pronounced just
u) actually originally had been latin "augustus". Something similar had happened to the Latin word
rubeus (red, reddish), which was transformed to modern
As for the Russian
рыжий, it indeed is related to
рудый which, in turn, is related to
There is also one thing we always should remember almost each time when we are saying that two words in different languages are related in deep historical context: those relations are not always straightforward. It's not always that simple that there existed some word in some mother language and that word had precisely that meaning and had been derived from exactly that word.
For example, the above mentioned Latin word
rubeus claimed to have had originated from
rubus (blackberry). And
rubus is related to
wr̥dʰo (some plant with red flowers, presumably sweetbrier or rose, or part of this plant). Oh, by the way, almost forgot to mention that
rose is also related to
рыжий, but, once again, not directly. And
wr̥dʰo is, once again, related to the first PIE root I've mentioned.