You can report this to the New York City Department of Health (here is a link) http://www.nyc.gov/html/doh/html/rii/index.shtml; However, with great respect, these types of accidents happen at even the finest restaurants. Further, if you suffered no injury or harm, you have no real claim, except perhaps for a refund for the price of the dish with the string in it. Also, be careful about accusing the restaurant of unsanitary or unhealthy conditions----restaurant employees make mistakes, but that does not justify a public accusation by you that the restaurant is unsanitary or unhealthy. You could be sued for defamation if your allegation is not true.
Based on the information you provided you were not injured by the string in your food. Thus, you would not have any claims against the restaurant. I agree with the other answer about contacting the health department.
DISCLAIMER: David J. McCormick is licensed to practice law in the State of Wisconsin and this answer is being provided for informational purposes only because the laws of your jurisdiction may differ. This answer based on general legal principles and is not intended for the purpose of providing specific legal advice or opinions. Under no circumstances does this answer constitute the establishment of an attorney-client relationship.
Without having been injured, you have no basis for a personal injury claim to be brought. As already advised, you can report the incident to the health department, who will likely direct an inspection to be conducted of the establishment. If you have eaten in this diner previously and have generally been satisfied with the food and cleanliness of the place, then you can presume that the ownership/managment are appropriately concerned and simply advising them of the incident will likely stir an internal investigation in the hopes of preventing a similar occurrence.