Winston acts cowardly in his relationship with Julia. If not for Julia's approaching Winston, he most likely would have never talked to her. Similarly, Winston is a coward when it comes to standing up for what he believes is the truth. He needs someone like O'Brien (who eventually betrays him) to spur him to take any action against the Party.
Later, when Winston is imprisoned in the Ministry of Love, he sees a woman who might be his mother, but he does not say anything to her because of his fear and apathy. Finally, Winston's most cowardly act is when he betrays Julia because of his fear of rats.
I in no way see Winston as a coward, I hate it when people try to portray him as a coward, he really was not! I think Winston was a boy (I think he would have allowed me to say that he was a boy, cause I'm now of the same age as he was, and I see myself pretty much as a girl)) pretty much in despair, he suffered from a bad health (varicose ulcer, and I think the book made it clear that he had also something painful in his shoulder) and despite that he also cared for his fellow human beings in that society, remember he helped a lot the Parson's woman, tasks that he did not like to do, (he did not like to use his hands) but that he did anyway, also the people around him turned easily towards him when they were in need for anything, to my opinion he was a really nice guy! He was a very thoughtful guy who was not willing to act easily, but that's only normal, he wanted to change society for the better but couldn't do that, society was too strong, by the way why should he have responded to the woman who said she might be his mother, in that society there might have been hundreds of mothers from the same age of Winston's mother, with the surname of Smith, meeting one of such a woman, did not oblige him to speak to her, I think he was really tired of his life, and also the same question why should he have bothered to speak to Julia, if she wasn't the one who made the first approach, he wasn't obliged to speak to her!