When is it necessary to say “_色的” when telling the color of something?
Ah..I'm not an expert in Chinese language but a native speaker, so the answer is just according to my experience. 

Very likely the character "色" could be omitted if you're about to use a 
word which its most usage is to state a color: white, red, yellow, etc. So the following are both grammatically good: 

1. 月亮是白色的 (the moon is of white color)
2. 月亮是白的 (the moon is white)


But when you want to be more accurate, the absence of "色" could cause ambiguity:

1. 剑是青铜色的 (sword is of bronze color)
2. 剑是青铜的 (sword is made of bronze)


What really cannot be omitted is "的". It is "的" that turn a noun into an adjective. We often use "X 是 Y 的" to tell "X is of the property of Y".

I could list some characters that in most cases their usage is to state a color: 

红 red
黄 yellow
蓝 blue
黑 black
灰 gray
白 white (also has the meaning in vain, but it's an adverb then)
紫 purple
绿 green