No, iOS devices can't use Flash and probably never will. Regardless of what you think about Flash, requiring it in order to access your website cuts out that part of your potential market.
Even if you argue in favor of Flash, it has a high overhead for your users. It slows them down, uses more resources and, even though it provides a "flashier" look, the overall user experience is worse, especially for mobile devices. (Apple's compatibility with Flash notwithstanding.)
If you want to track your visitors' progress through your site, which pictures or pages they view and how they access different parts of your site, Flash will make that more difficult. It can also be a dead end for search engines so, if you want to get your website ranked in Google's or Yahoo's search results, Flash will make it harder.
Flash is also a good way for users to get a virus. There are many XSS (cross-site scripting) attacks which make Flash vulnerable. In order for Adobe to patch all these holes, they would have to make all previous versions of Flash player/plugin unusable. Consequently, I and many other users set up their browsers to automatically block Flash content, rendering your site invisible. I believe this is why 131802's computer came up with a blank page. His Linux machine either blocks or does not recognize Flash.
I can get by when they visit a website that has some Flash content scattered throughout but, I really don't recommend building a whole website out of Flash.