I am in the process of writing a novel, and I'm just trying to get some ideas and thought on this. I am not sure yet if I want to completely remove all dialogue, but at this point I feel that dialogue would wreck the characters and the image I am trying to convey. But I am just curious to see others' thoughts on a novel with multiple characters but no dialogue at all.
Short answer: yes.
The question is complicated, though, by what counts as 'dialogue'. If you read - for example - Birgit Vanderbeke's The Mussel Feast (Peirene Press, 2013, trans. Jamie Bullock) you won't find any direct speech - i.e., dialogue in quotation marks. But there's plenty of reported speech in there. It's a more difficult technique than direct speech, but the reward is that the entire texture of the prose is conditioned by the narrative voice. There's a greater cohesion, something a little more like storytelling and less like conventional prose fiction. I bring this up just an example, to show that the question has more nuance than just dialogue vs. no dialogue. It's a question primarily of form, and secondarily of content. You haven't gone into much detail about the aesthetic considerations in play, so I can't offer any advice specific to your piece.
And as for writing a novel that most people would like (mentioned in another answer this this question). Screw that. Write a novel you think is the best novel you can write, and would be enjoyed by a reader with precisely your taste. If your taste coincides with the zeitgeist, so much the better. But if you're not writing for yourself, you will be able to smell your novel's inauthenticity from the other end of the block. (The other, complementary piece of advice is to read both broadly and deep, so your taste is as informed as it can be .... but that's a given.)
Is it possible? Yes. Actually you could write it right now.
But, I think it's the same as asking yourself:
Is it possible to make a movie with no sound?
Can it be enjoyable to date a girl/boy who doesn't talk?
Yes. Maybe. For some. But I'm pretty sure most people won't like it.
So the question actually is: Do you want to write a novel that MOST people would like?
If the answer is no, well, then go on.
(PS: If you write the novel and many people like it, then I retract what I said.)
E.L. Doctorow's 1975 novel Ragtime had no dialogue in it, although it did have descriptions of conversations. The book did not contain any dialogue in "quotes." It's a widely acclaimed novel, and has even been turned into a film (released in 1981, directed by Milos "One Flew Over the Cuckoo's Nest" Forman) and a Broadway musical. So it is possible. But Doctorow is a damn good storyteller, which helps.
The following English language novels have no dialogue:
- Journey to the End of the Millennium by Abraham B. Yehoshua
- The Secret Diary of Adrian Mole, Aged 13¾ by Sue Townsend (turned into a tv series)
- The Making of Americans by Gertrude Stein
The following German novels have no dialogue:
- Exerzierplatz (The Training Ground) by Sigfried Lenz
- Das Treffen in Telgte (The Meeting at Telgte) by Günter Grass
- Die Herrlichkeit des Lebens by Michael Kumpfmüller
- Der Vorleser (The Reader) by Bernhard Schlink (turned into a movie)
- Gut gegen Nordwind by Daniel Glattauer (This novel consists of the email conversations of a woman with a man who erroneously received her first email, the cancellation of a magazine subscription, due to a typo in the email address. The novel sells well, ranking 642 in books on amazon.de.)
The following French novels have no dialogue:
- À rebours (Against Nature / Against the Grain) by Joris-Karl Huysmans (a classic)
- La petite robe de Paul (Paul's Little Dress) by Philippe Grimbert
- Les Derniers Jours de Stefan Zweig by Laurent Seksik
- Lame de fond by Linda Lê
- La nuit zoologique by Claude Durand
I'm sure there are more.
I think it would be possible but a bit boring. Furthermore your readers want information of how your characters would respond. And no one would know, how they get along with each other. The dialogues between the different characters are very important. By the words they say, the readers will know how they relate to each other.
Besides, the dialogues make certain situations more interesting and give them a bit variety. It makes everything natural. In addition your characters would get more personality.
But I have to say that I like such ideas. They are special and trying something new is always good. Make your experience.