Is a nation giving up sovereignty if it joins the International Court system?
There are a number of ways in which to answer this question, depending on how you conceive of sovereignty.
In a sense, a state is not giving up any sovereignty when it joins the International Court system. Looking at the International Court of Justice, for example, we see that no state may be placed under the jurisdiction of the court without its (the state’s) consent. In other words, the state does not give up its sovereignty because it is voluntarily choosing to submit to the court.
However, in another sense, states do give up their sovereignty. If a state signs a treaty putting itself under the jurisdiction of the court, it is no longer completely sovereign. Now, a body that is not within its own territory and not subject to its laws can order it to do things. This means that it has lost the ability to be completely sovereign.
Thus, depending on how we define sovereignty, the answer to this can vary. A state does submit itself to the authority of an outside body. However, it does so willingly. People can differ as to whether this constitutes a loss of sovereignty. I would argue that it does constitute such a loss because the state is no longer completely free from outside control.