No.
They cared deeply about each other as friends, but there was never anything romantic in it.
Holmes was asexual. He wasn't just uninterested in women, he was uninterested in romance.
All emotions, and [love] particularly, were abhorrent to his cold, precise but admirably balanced mind. He was, I take it, the most perfect reasoning and observing machine that the world has seen, but as a lover he would have placed himself in a false position. He never spoke of the softer passions, save with a gibe and a sneer. They were admirable things for the observer--excellent for drawing the veil from men's motives and actions. But for the trained teasoner to admit such intrusions into his own delicate and finely adjusted temperament was to introduce a distracting factor which might throw a doubt upon all his mental results. Grit in a sensitive instrument, or a crack in one of his own high-power lenses, would not be more disturbing than a strong emotion in a nature such as his.
-- "A Scandal in Bohemia"
"[...] love is an emotional thing, and whatever is emotional is opposed to that true cold reason which I place above all things. I should never marry myself, lest I bias my judgment."
-- The Sign of Four, Chapter 12
Watson was heterosexual, and indeed married during much of his friendship with Holmes.
As I listened to the words and realised what they meant, a great shadow seemed to pass from my soul. I did not know how this Agra treasure had weighed me down until now that it was finally removed. It was selfish, no doubt, disloyal, wrong, but I could realise nothing save that the golden barrier was gone from between us.
"Thank God!" I ejaculated from my very heart.
She looked at me with a quick, questioning smile.
"Why do you say that?" she asked.
"Because you are within my reach again," I said, taking her hand. She did not withdraw it. "Because I love you, Mary, as truly as ever a man loved a woman. Because this treasure, these riches, sealed my lips. Now that they are gone I can tell you how I love you. That is why I said, 'Thank God'."
"Then I say 'Thank God', too," she whispered as I drew her to my side.
Whoever had lost a treasure, I knew that night that I had gained one.
-- The Sign of Four, Chapter 11
"Now, Watson, the fair sex is your department," said Holmes, with a smile, when the dwindling frou-frou of skirts had ended in the slam of the front door.
-- "The Adventure of the Second Stain"
Watson frequently encouraged Holmes to show an interest in women, including while they were alone - a thing he would hardly have done if they were intended to be a homosexual couple.
I also found a very interesting, lengthy, and rather heartfelt article from the Sherlockian Sherlock fan site, which goes into this issue in some detail.
Guessing the sleuth’s sexual preference is not a novel thing, despite of the fact that in the original Canon dr. Watson is married three times, and even Holmes remarks in The Second Stain that his friend knows a lot about women („Now, Watson, the fair sex is your department.”) [...]
Roger Johnson, editor of the Sherlock Holmes Journal, observed that in Arthur Conan Doyle’s original stories "Watson is something of a ladies' man, but a faithful husband to his wife. And Holmes is essentially asexual, with no erotic interest in women or men." [...]
Doyle’s detective was not homosexual, and he did not changed this later, when he was rich and famous, though he could have written stories under a pen name. He was not too interested in erotic themes, but love between men and women often appears in his works. With the figure of Sherlock Holmes he wanted to focus on spiritual values. Some may think it odd that Holmes does not have any romantic relationships, but he explains in the Canon that thinking and his profession are the most appreciated things in his life, and he thinks emotions like love would only withdraw him from them. [...]
Legendary Sherlock Holmes actor Jeremy Brett, who lived together with a male partner for a while, said that there was a genuine friendship between Holmes and Watson, but nothing more. Sadly in the modern age relationships changed so enormously that most people simply cannot understand how such kind of friendship can exist. Jeremy Brett had a humane and fair attitude to the topic. It was clear to him that Doyle did not want gay characters, and he respected this. In his interviews he always emphasized that the relationship of the duo was devoid of any romance, what they felt for each other was honest friendship.
