Nope, NI and NHS are separate things.
Everyone who is eligible for NHS treatment can register with the NHS (and will be given an NHS number), but you only need an NI number if you are going to be working in the UK.
For example, all British people get registered with the NHS and given an NHS number when they are born, however, they will not be issued an NI number until they turn 16 and are legally allowed to work full time. I was born and raised in the UK, but didn't have a National Insurance number until I turned 16 (in 1999).