According to Chris Dixon, Twitter and Facebook now disallow Google to Index any of their content.

Last year and in the past, plenty of Interesting experiments have been conducted with your question as the basis. See a few of them here:

How do Tweets Influence Search Rankings? An Experiment For a Cause | SeoMoz
Exactly How Powerful are Tweets and Retweets. Help us Find Out | SeoMoz

As of now, it's clear that "inbound signals" thanks to your Tweets affect search rankings. They can help you gain traction, get traffic, and help your website rank better.

As to the direct indexing of Tweets, it's a "no".