Well it really comes down to whether healthcare is a human right or not.
Right now no one, immigrant or otherwise, is turned away if they have an emergency healthcare issue in the US. So we already provide universal emergency healthcare, no matter how you slice the population up, and that has existed since the 80's. In some places illegal immigrants find that if they seek healthcare they may get a visit from INS, but that doesn't prevent them from receiving emergency care at the time that they need it. Being able to receive healthcare doesn't also guarantee that you are immune from following the laws of the nation, but that shouldn't be surprising, hospital workers are required to report gunshot wounds to the police. Healthcare doesn't need to be anonymous to fit the rule of universal.
That doesn't mean that we provide comprehensive health maintenance to everyone, though.
So at what point is it a human right? Just for life saving? For regular check ups, cancer care, etc? For elective medication like birth control?
It's that question w are struggling with as a nation, and providing truly universal emergency healthcare to everyone, and adding comprehensive health maintenance to over 90% of the citizens with 98% or more coverage to those unable to afford it themselves isn't a small thing.
It may still fall short of the goals some want to pursue, but providing healthcare to those who are here illegally and have no legal right to be inside the country, never mind benefiting from public social programs is going to be a contentious issue for a long time to come. Many people don't believe comprehensive healthcare is a human right, and that emergency healthcare is far enough.