To clear up a common misconception, the congratulations note Sasuke sent to Sakura on the day of Naruto's wedding was not a birthday message. I don't know where the myth that the congratulations was in regards to Sakura's birthday emerged from, since we have no idea what date Naruto and Hinata's wedding took place other than that it transpired sometime during the spring. We have no clue if the wedding took place on Sakura's birthday or not. Since the anime doesn't inform us viewers, I assume not. It's a wedding arc, not a birthday arc, and we the audience all have matrimony on our minds; of course the note was meant to congratulate Naruto on his marriage to Hinata unless directly stated otherwise, which it wasn't.
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Besides, who says "congratulations" for a birthday? I realize that it's an English translation and there may be a Japanese cultural detail I've missed. But in English, people wish you "happy birthday", not "congratulations". Congratulations is what you say to a pair of newlyweds upon their marriage.
So then why was the note delivered to Sakura, not to Naruto, its intended recipient? Some fans have mentioned that in Japanese culture, couples deliver wedding congratulations to the bride and groom together. By sending the note of congratulations to Sakura, Sasuke's letter has layered meaning. The first is obvious: he is happy for Naruto and wishes the newly married Naruto and Hinata congratulations on their marriage and new life together. The second meaning is less obvious, another subtle way of Sasuke expressing his feelings for Sakura: he's saying that they are a couple, since Sakura will give his note to Naruto, thereby it's like he and Sakura congratulating the newlyweds together as a couple themselves.
To clarify, I'm not that knowledgeable on Japanese culture, so I haven't been able to verify this explanation myself, it's only an explanation I've heard from fellow SasuSaku fans. Though considering how Sakura blushes and thanks Sasuke for the wedding congratulations, and how Sasuke stops and smiles happily on his travels, this cultural explanation seems like the best explanation, even if it is only headcanon. Otherwise Sasuke congratulating Naruto and Hinata for their wedding doesn't seem like cause for Sakura to blush. SasuSaku fans' explanation that Sasuke is insinuating that he and Sakura are a couple (and thus purposefully sending the hawk to Sakura instead of directly to Naruto) is the best explanation I've heard so far, and it fits with how at that wedding scene all the couples (InoSai, ShikaTema) were being shown, and it transitions to showing SasuSaku as well.
