Juliet writes to Mark and says that she didn't refuse, she just wanted time to think—but says that Mark was too busy ranting about Sidney and Guernsey to notice. She points out that they've only known each other for two months and after almost marrying a man once before, she's not willing to make a mistake again. She insists they don't actually know each other well enough to be married.
Notice that Juliet refers to marrying Mark as a mistake already—this tells the reader that Juliet will go on to say no, even if she dithers in the moment. It's also clear that Mark is jealous of Juliet's life that doesn't include him, which means he's jealous of her work in particular.
Juliet writes to Sophie about her proposal. She says that Mark offered her a huge diamond ring and is very upset that she didn't say yes straight away. He became convinced that Juliet was actually in love with Sidney. Juliet says they were at his flat by then, and Mark started yelling about Sidney, Guernsey, and pen pals. Finally, Juliet started to cry, which made Mark feel bad. Juliet almost agreed to marry him, but thought about spending her life crying in order to make him treat her kindly and decided not to. They fought for a few hours and as Mark helped her into the car to go home, he called her an idiot.
Mark recognizes that he's a desirable bachelor: he's rich, powerful, and leads a glamorous life. He knows that for plenty of women who value security above all else, he' be able to offer them everything the want. Juliet, however, is more interested in her work (Guernsey, in Mark's rant) and her life outside of a possible marriage than she is in the kind of security Mark could offer, hence her decision to say no.
Juliet suggests that maybe Mark is right. She asks if Sophie remembers the romance novels they used to read as girls and one of the dashing men from those novels. Juliet says that Mark is just like that character: tall, handsome, and other women talk about him all the time. Juliet writes that she has a nagging feeling that he's not right for her, especially since he's trying to keep her from going to Guernsey to write a book about the Occupation.
Juliet's wondering if Mark is right that she's an idiot speaks to the power of the cultural narratives of the era: Juliet understands that she doesn't want what's considered normal and acceptable and is therefore turning down an acceptable life path in favor of unknowns, but freedom nonetheless.