For the most part, I think that his conception of organic solidarity does apply to our modern society. However, I think that it overstates to some degree the extent to which we rely on interdependence to make us feel solidarity with one another.
It is certainly true that we do not have a system that is based so strongly on everyone having the same values as we once did. The United States is too fragmented in terms of ethnicity and also of cultural belief for this to be true. In that sense, Durkheim's view of organic solidarity seems accurate.
But when you consider the larger picture, it is clear that some of what binds us together is our common beliefs. For example, it is clear today that we need China. They make many things that we do not and so we are interdependent. Yet there is no sense of solidarity with China. Instead, we feel solidarity with other Americans, whether we actually need them or not. This is presumably due in large part to our shared beliefs. We no longer share detailed beliefs about things like religion, but we continue to share a general belief in things like democracy, personal freedom, and the rule of law. We identify ourselves with others (whether in the US or out of it) who believe in such things.
Therefore, I cannot agree completely that we live in a society that corresponds to Durkheim's idea of organic solidarity.