Hi Karl,
First thing is first: you don't necessarily need to stop play unless the person who has run out onto the pitch has somehow become involved in active play. Given that you feel that this person will be on the field for an extended period of time, however, and he is there to treat an injured player, I don't mind that you stopped it up right away -- but be careful, if the other team is going to go on a quick counter attack, they might be rightly pissed that this has happened.
Second thing, the correct restart for when a team official enters the field of play and interferes is a dropped ball from the position of the ball when play was stopped. Likely, you will drop the ball and allow the GK to pick it back up.
Third thing, unless the physio is a player, a substitute, or a substituted player, he cannot be cautioned. The correct procedure is to have him removed from the FOP (that is, returned to the technical area) and if his behavior is irresponsible, he should be removed from the immediate surroundings as well. I would suggest that if, in your opinion, he has come on knowing fully that he cannot come on without your permission in an effort to stop the opposition from a quick break and therefore his decision to come on was tactical, then he is guilty of irresponsible behavior and should be removed. Likewise, if he starts yelling and swearing at you for making him leave the FOP, he needs to be removed as well. However if it was an honest mistake, and he leaves as soon as you ask him to (although at that point, might as well let him treat the player), then no need to eject him. I would, however, mention at half time or full time that you cannot enter the FOP without the referees' permission.
Here is the relevant section from the LOTG, page 64:
Team officials
The coach and other officials indicated on the team list (with the exception of
players or substitutes) are deemed to be team officials.
If a team official enters the field of play:
• the referee must stop play (although not immediately if the team official
does not interfere with play or if the advantage can be applied)
• the referee must have him removed from thefield of play and if his
behaviour is irresponsible, the referee must expel him from the
field of play and its immediate surroundings
• if the referee stops the match, he must restart play with a dropped ball
from the position of the ball when the match was stopped, unless play was
stopped inside the goal area, in which case the referee drops the ball on
the goal area line parallel to the goal line at the point nearest to where the
ball was located when play was stopped
