There is, unfortunately, no evidence for this actually being a thing in Ancient Greece (and I didn’t realise this was a thing on the internet either until looking it up now). This urban legend (or whatever it is) perhaps derives from the fact that apples were closely associated with the goddess Aphrodite, i.e. with love. Why this is the case is impossible to answer: trying to assess why certain animals or fruits or trees became associated with certain gods takes you to increasingly speculative places. The fact that the Greek word for apples (μῆλα) could be used as a euphemism for ‘breasts’, as at Aristophanes Lysistrata 155-6 ("Menelaus once saw Helen’s naked apples, and chucked away his sword") and [Theocritus] Idyll 27.50 (“I’ll 'teach' your apples first”), is probably a coincidence.

Nonetheless, there are a few stories from Greek mythology in which an apple was used to seduce a woman (not really proposing). The most famous these days will be Paris and the golden apple, which he gave to Aphrodite in exchange for her helping him seduce Helen (Aristophanes is presumably making a double joke in the Lysistrata: Paris gave an apple to get Helen, and then Helen gave her ‘apples’ to persuade Menelaus not to kill her). But then, the golden apple could have been given to any of the three goddesses, and it’s an indirect exchange, so it’s not a particularly close fit.

The story of Atalanta and Hippomenes is closer to the idea of apples being used for seduction. It’s narrated by Ovid (through the figure of Venus) at Metamorphoses 10.560-680: Atalanta was a young woman who was given a prophecy by an oracle that getting married would turn out weirdly for her, so decided that she would only marry someone who could beat her in a race, and kill everyone who lost. A guy named Hippomenes comes along and challenges her, and because she swore an oath that this is what she would do, she accepts his challenge. Ovid, however, says that she’s already in love with him, and is reluctant to do the race (10.614 “I’m not touched by his beauty … but I could be touched by it”, 10.629-30 “If only you were happy to stop, or – since you’re out of your mind – that you were faster”). Hippomenes manages to cheat the race and her oath simultaneously by throwing some golden apples that Venus gives him off to the side, and she – because she likes gold/doesn’t want to kill him – runs off to pick them up, allowing him to win. Then they have sex in a cave and turn into lions. So the apples here aren’t exactly seducing Atalanta, as much as being associated with the seduction of her, and making it possible for them to marry. It’s a bit more direct than with Paris and Helen, but still a little off.

By far the closest myth that connects an apple to love/marriage is one that is (I think) almost completely unknown today: Acontius and Cydippe. It’s told by the Hellenistic Greek poet Callimachus in his Aetia (fragments 67-75), and was insanely popular and influential on the later Roman poets (and subsequently any other poet who has drawn on Catullus, Virgil, Ovid etc.). The story is this: the two were both incredibly beautiful, and Acontius saw Cydippe at a festival and immediately fell in love; because just going and chatting up random girls at festivals was not super respectable behaviour, and because Cydippe had been betrothed by her dad to someone else, Acontius was taught a trick by Eros: he threw a golden apple in front of Cydippe, on which he wrote “I swear by Artemis to marry Acontius”. She picked up the apple and then read out the words, at which point boom, oath sworn, no escape. Before she can get married to her fiancé, she falls mysteriously sick, so the wedding is delayed; the next time it’s due to take place, she falls ill again; the next time it’s due to take place, she falls ill again; the next time it’s due to take place, she falls ill again. This time her dad figures something might be wrong, so heads off to Delphi to ask Apollo what’s happening. The god replies that Cydippe swore an oath to Artemis that she would marry Acontius, and so she has to do that, and also that Acontius is a real catch anyway (“you will not be mixing lead with silver, but electrum with shining gold”). So they get married and live happily ever after. So in this case there is a really strong connection between the apple and the marriage, but he’s still not really proposing to her – it’s more like he’s getting her to propose to him.

So no, neither men nor women in ancient Greece would propose by lobbing apples at people. But the fruit was closely connected to Venus, and in at least one myth someone did manage to marry the girl of his dreams (as ever in the Greek world, she did not get much of a say in this) by chucking an apple in her general direction. And that was genuinely a love story for the ages.