Well, once tissues leave our bodies they aren't technically a part of us anymore, in a physical sense. They do, however, contain our DNA.
I think the reason the Supreme Court came to that decision is that the scientists and doctors that collect those tissues will usually have the people's consent. The tissues they use, we got rid of. They are like rats looking through the trash. Say you got your appendix removed, then a doctor could use it for scientific research. You don't need it anymore, it is no longer a part of you because it is outside your body.
I also think it would be really complicated if everyone started claiming ownership over every single thing out in the market already. Could you imagine the political and economical fiasco? I agree with the ruling, although doctors should inform the patient before hand if they plan to use their tissues and cells.
It is important to note that HeLa cells were taken without Henrietta Lacks' or her family's consent and it just shows how important it is to ask and inform first.