The answer is yes.
Medical malpractice trials, accident cases, and wrongful death cases are civil lawsuits. The courtrooms for all those cases just listed are always open to the public. That means if you are walking around the courthouse, and want to sit in and listen to what is going on during a trial, you simply open the door walk in and sit down.
You can stay for as long or as short a time as you wish. You can watch the actual workings of any civil lawsuit on trial in New York.
There may be one exception.
At the end of a trial, when the judge is giving the jury legal instructions on the law, the judge will usually order the courtroom to be locked during the time he's giving the jury instructions. This is not done to prevent the public from coming in and listening or watching. Instead, it is simply done to prevent distractions from occurring so that the jury pays careful attention to the legal instructions.
You do not need identification to walk into a courthouse, you only need go through security and then walk the halls to find a courtroom with activity.
I highly encourage anyone interested to watch a real trial to go to their closest County courthouse and sit in for a while. It's an eye-opening, fascinating look at how the lawsuit process works.