Before the ball enters the goal from an attacking player’s shot, a spectator enters the field of play and slightly touches the ball with his hand but does not manage to stop the goal. What decision should the referee make?

Answer (November 15, 2015):
In such cases, the referee must follow the guidance on p. 66 of the Laws of the Game:

Outside agents
Anyone not indicated on the team list as a player, substitute or team official is deemed to be an outside agent, as is a player who has been sent off.
If an outside agent enters the field of play:
• the referee must stop play (although not immediately if the outside agent does not interfere with play)
• the referee must have him removed 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

In your situation, Law 3 requires that the referee determine whether or not the outside agent—here the spectator—has truly interfered with play. Only the referee on the game can determine this; not the players, not the team officials, no one but the referee, with advice from the ARs, if necessary.