NO. many physicians are in private practice, others are independent contractors in hospitals, clinics, nursing homes, etc.. , there is no reason for them to form a union.
Those who are employed by hospitals and other businesses are discouraged from forming unions, just as all other employees who work for them are.
The government has little control over doctors other than if they should attempt to fix prices then they would be subject to government penalties under anti trust laws.
Your last statement regarding "the fed gov leading the way in failing to pay...." is simply not true. This kind of rhetoric is from the right wing conservatives to try and discredit what they consider 'entitlement programs'. There is so much disinformation being spread around, one hardly knows what is truth anymore.