And believe me, Europe has nothing on South Africa's constitution right now.
It's not just about the constitution, its about accountabiity.
The US probably has great guarantees of human rights in the consttution, however they are useess when there is noone to actually protect them and hold violators accountable. The Supreme COurt is a quasipolitical body that is always influenced by politics and the administration when deciding, for example, what is constitutional and what is not. There is no higher body to say a state has stepped out of line or violated the rights of it's citizens.
And that's because the US has always tried to distance itself from international law, and any treaty that can hold the US accountable.
The US as not ratified the Inter-American treaty of human rights, they have not signed Optional Protocol One to the ICCPR which allows individuals to submit claims of human rights violations to the Human RIghts COmmittee. The US has not ratified the Rome statute of the International Criminal Court either. Not a great way to lead by example.
So, basically, if you think your human rights have been violated and the courts in the US disagree, you're basically screwed, there's noone you can turn to.
Things are different in Europe. All european countries are members of the COuncil of Europe and have signed the European convention on human rights and are subject to the authority of the European COurt for Human Rights, where individuals can submit claims for violations of their human rights by the state, and the Court can find the state liable.
Also, in the EU you have the European Charter of Fundamental Rights which prvides a wide range of rights to be protected, including many second generation rights.
THerefore, human rights protection in Europe is far ahead of any other states in the world.