Not all mysteries have character arcs. (I would guess that most do not.) The detective comes to a conclusion, but he/she isn’t likely to realize a flaw and find a way to grow. Many SF stories are like that, too. A lot of romances are about the characters getting past misimpressions and discovering they are made for each other. Usually, one character doesn’t change at all. Often neither does. One of my favorites, The African Queen, has both characters with big changes. She is narrow and intolerant. He’s an alcoholic with few principles. They both grow significantly. They change each other as they face perils together. Casablanca’s Rick has an amazing character arc (while Ilsa does not).
So, it is perfectly acceptable to write a romance without a character arc. Discovery, rather than making an important internal change is welcomed by many readers, Rather than seeing a struggle to grow, people are entertained by clever turns on the tropes, misimpression’s (often leading to humor), the conniving of villains, erotic moments, intriguing premises, wish fulfillment, and fun settings. All good.
Having a character grow is a choice the author makes. Many romance writers want to include growth. Many don’t. If there is such growth, there is a character arc. How steep the arc is depends on how serious the protagonist’s flaw is. (If the character has no flaws, there cannot be an arc. There’s no place to go.) The steeper the arc, the more worried readers are likely to be about the protagonists because such change is risky and painful. Worried readers (unless driven away) are engaged readers.
A change in perspective can be enough. There’s only a minor character arc when a protagonist decides a style of music (jazz, country western, rock) isn’t all bad. There is a huge character arc when a hero or heroine forsakes lifelong prejudices. Huck Finn famously rejects slavery even though that institution is accepted and defended by everyone he knows. That’s a heck of an arc.
Most stories with character arcs include external goals. That’s because the quest of the character isn’t usually to become a better person. They either accept their flaws as unchangeable or don’t truly realize how serious they are. It is in facing the obstacles to the external goal (and struggling with them) that the character understands his or her flaw(s) and becomes motivated to change.