Would Ryuk have died if he had killed the SPK? In the manga, Light begs Ryuk, to kill everyone in that room. So let's say, Ryuk would have killed the Japanese Task Force and the SPK. Would Ryuk die after he killed them?
Rule LVIII:

1) By manipulating the death of a human that has influence over another human's life, that human's original life span can sometimes be lengthened.

2) If a god of death intentionally does the above manipulation to effectively lengthen a human's life span, the god of death will die, but even if a human does the same, the human will not die.

This seems pretty explicit to me.  Note that this seems to refer to killing with the death note but if a god of death kills by any other means he is given an "Extreme Level" punishment and then is executed.

What's love got to do with it? Nothing