Would Ryuk have died if he had killed the SPK?
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