[T]he hero of my story, whom I love with all my heart and soul, whom I have attempted to portray in all his beauty and who has always been, is now, and will always be supremely magnificent, is truth.
--Tolstoy, "Sevastopol in May", collected in The Cossacks and Other Stories, Penguin Classics (David McDuff, trans.), p. 255
I think "proto-modern" is appropriate. In general, the literary modernists of the early 20th Century -- Woolf, Joyce, Eliot, Proust, et al -- were heavily influenced by the literary realism of the 19th century -- including not just Tolstoy but also Flaubert, Dostoyevsky, Stendhal, and others. In some ways the modernists were carrying on the realists' work, and in other ways they were rebelling against it.
Many of the defining works of modernism, such as Ulysses, Mrs. Dalloway, and In Search of Lost Time were attempts to portray characters' lives truthfully and comprehensively. Tolstoy had made truthful portrayal of the world his primary goal over his 60-year career, and Anna Karenina in particular, with its detailed reproduction of life in 1870s Russia, was a clear example to the modernists of what had been done before. And yes, Tolstoy's use of stream of consciousness techniques* was certainly noted, and built upon. (One of the ways Joyce et al. one-upped him was by committing to stream of consciousness as the primary form of narration -- after all, if you're really going to describe a person completely, you must show their world from their viewpoint. In Tolstoy you find both the stream of consciousness and a less intimate way of representing characters' thoughts)
As for specific connections between modernists and Tolstoy:
- Joyce called him a "magnificent writer" who is "head and shoulders above" others.
- Woolf wrote a review of one of Tolstoy's early works for the Times Literary Supplement, in which she particularly noted his narratorial felicity in moving between characters' points of view.
- Hemingway considered Tolstoy one of the greats he had to "beat" in order to become a truly great writer.
- Nabokov enthusiastically lectured about Tolstoy to his college students, and even "donated" his own analysis of Tolstoy's use of timelines in Anna Karenina to the main character of Pnin.
*Tolstoy was not the first writer to use stream of consciousness -- earlier examples include Sterne's Tristam Shandy and Poe's "The Tell-Tale Heart." But his works in general, and the ending of Anna Karenina in particular, are notable early examples of the style.