I don't know for sure, but it may depend on the pigeon. Most animals that are healthy, and know where/how to get the food, will not starve themselves. But there can be many reasons for a specific animal to not eat, even when the person thinks it should.
If the pigeon does not recognize that you are offering food, it might starve without understanding that it could get food by eating from your hand. Putting the food in a dish, and refilling the dish every day, gives the pigeon a chance to learn that stuff is food, and that you bring food.
The pigeon might decide it is hungry enough to eat from your hand-- at a time when you are not present. (I assume you are not spending ALL day sitting there offering food in your hand.)
Even if a pigeon would start to eat from your hand before it starves to death, it might go hungry long enough to cause health issues.
How long has the pigeon been without food so far? A few hours should be fine, but several days would have me worried. (I don't have pigeon-specific knowledge, just going by times that are fine or problematic for some other animals.)
