Can you read the great books on a Kindle? Most likely, yes.
However, a word of warning. All the links on this site point to paperback editions. If you're interested in Kindle editions of a particular book, be careful to ensure that the translators are the same. At the time of writing this guide, Amazon doesn't seem to care about consistency of translations across the physical and e-book versions. So a link to a paperback version of a book translated by a particular translator may link to a Kindle version translated by a completely different person.
On another note, for more serious reading, the physical books are probably preferable if you can afford the time and space needed. I say this as someone who loves his Kindle and carries it my pocket everywhere I go.
The first problem I've encountered is that scribbling notes is difficult on the Kindle, which makes for significantly lower retention. Somehow, the act of writing seems to force my brain to retain things much better, even if I never look the notes again. So if you do want to read on the Kindle, I would suggest making sure you have a notebook and pencil at hand to write down your thoughts as you read.
The second problem with the Kindle is that navigation is quite terrible. This is not a criticism of the Kindle specifically, so much as against the current state of e-ink technology. This wouldn't be so much of a problem if all the available books were formatted nicely for the device. Unfortunately, the reality I've come across is that many books don't correctly incorporate footnotes, endnotes, and table of contents. This makes it very hard to use the Kindle on books where you have to constantly jump between the text and endnotes, or constantly read footnotes in that little popup. Jumping through the chapters or sections of a book in order to skim it is hard to do for many books that don't correctly generate a table of contents. You'll also come across many transcribing errors for older books. Finally, many of the books on the list don't have Kindle editions available.
So yes, you can read these books on the Kindle for the most part. However, if possible, I recommend buying the paperback editions. They're just generally more amenable to serious reading. Besides, building up a beautiful personal library of incredible books is a journey of its own!