I'm going to have to disagree with my colleagues unfortunately, if only academically. You can file a petition for rehearing of the denial of certiorari within 25 days, under Rule 44(2) of the Supreme Court Rules of Practice. The grounds of the rehearing are limited to intervening circumstances of a substantial or controlling effect or to other substantial grounds not previously presented, and accompanied by a certificate to that effect.
No. However, if this is a criminal case and the United States Supreme Court has denied certiorari on direct appeal, it might be possible to pursue a collateral challenge to the conviction in the original trial court, assuming that you have a factual and legal basis for doing so. Understand, however, that at each step in the process (pre-trial negotiation, trial, appeal, post-conviction, habeas corpus, etc etc) the probability of success grows smaller and smaller and pursuing all possible avenues of relief can become a pursuit of shadows.