There are many people who cook vegetables without washing them (before cutting) because they think it removes important nutrients.
This claim is predominant in some Asian countries.
Is that true?
Yes, washing vegetables does remove some of the water soluble vitamins, specifically B-complex group, and vitamin C; however, in general, the health benefits of washing vegetables by avoiding pathogens as well as residual pesticides, herbicides, and other such residuals greatly outweighs the nutrient loss.
With regards to the nutrient loss, vitamin B-12 can be loss through the washing process; however, generally this is only a concern for vegans who are unable to get the vitamin through animal sources. For example, most of the nutritional value of rice is lost during milling which requires that it be enriched prior to sale. As the enrichment is on the surface of the rice, it can be lost if the water is discarded. Furthermore, the degree of vitamin loss can be mitigated by using proper washing and handling techniques and by limiting the washing of the vegetables until they are ready to be used and limiting the amount of washing that is needed (i.e. once washed, avoid contact with dirty vegetables; wash vegetables before they are chopped, etc.).
Barring the issue of vitamin B-12 and vegans who generally use supplements to resolve the problem; the nutrient losses are greatly offset by the risk of food poisoning caused by a not insignificant list of food-borne pathogens.
Washing vegetables before you cut them wouldn't remove any nutrients since the skin of the vegetable protects it, just like in rain outside etc. Leaving vegetables soaked in water for to long and the protecting layer would disolve and the water get inside the vegetable though. Cutting the vegetables and then soaking them in water could also very likely, depending on the vegetable in question, remove nutrients.
U.S. Department of Agriculture recommends that you wash the vegetables to get rid of dirt and pathogenic microorganisms. I would find that a strong case to wash ;).
The claim appears to be "It is healthier not to wash vegetables at all, because washing may remove some of the nutrients."
This claim only considers one potential benefit of unwashed vegetables. It ignores a number of potential downsides, including potential leaving on the vegetables microorganisms known to be harmful (E. coli, Salmonella, Clostridium, Bacillus, Listeria), and also pesticide residues which are potentially harmful.
Even if it were true that some nutrients were lost, the risks of food poisoning and death outweigh the benefits.
In my searching, I found experts from the following science-based health organisations recommend washing your vegetables to remove microorganisms and contaminants:
I found more articles from the following non-peer-reviewed media organisations that also recommend washing your vegetables:
Even Oprah, Princess of Woo!
I found the following articles from any source at all recommending you not wash your vegetables to preserve nutrients:
Now, I wouldn't argue that the most popular point of view is necessarily the correct one, but the importance of washing vegetables compared to a claim about nutrients (an alleged claim, at that) isn't in the slightest bit controversial.