Just to confirm - since 18 April this year the EU Directive on public procurement requires clinical health services contracts over around £600k for the NHS to be opened up to competition. Most of these contracts will be won by NHS Providers of course, who are often best placed and will win the competition on objective criteria. However the way is open for private providers, e.g. Virgin Care, to enter these competitions and in some cases to win these.
This replaces the position we had under the Health and Social Care Act 2012 (which was a New Lab concept passed under the ConDem coalition) - where commissioners of clinical health services were required to consider whether holding a competition was in the public interest and the best route to securing a good deal, but it didn't necessarily oblige them to do this. Often competitions were held and private sector providers have been appointed.
It is true to say that the EU laws on this make it more likely that we will see an increased number of private providers in the NHS (though this does depend on whether they can and want to compete with NHS providers, who are often very well placed to deliver the services).
Its also true to say that if we left the EU then a left wing government could remove those elements which require commissioners to compete contracts and allow them to just give the contract to the NHS provider down the road.
As a remain voter, who doesn't want to see privatisation of the NHS, this has been the biggest issue for me to get over.
On the other hand the biggest issues facing the NHS are funding and staffing and I believe to deal with these we need free movement of people and we also need economic certainty (and the tax receipts to fund the NHS that go with it).
That's why I voted remain (together with all sorts of other reasons).