Choice C is the best answer. Akira states that his unexpected meeting with Chie occurred only because of a "matter of urgency," which he explains as "an opportunity to go to America, as dentist for Seattle's Japanese community" (lines 41-42,"I've... community"). Akira decides to directly speak to Chie because Chie's response to his marriage proposal affects whether Akira accepts the job offer.
Choice A is incorrect because there is no evidence in the passage that Akira is worried his parents will not approve of Naomi. Choice B is incorrect because Akira has "an understanding" with Naomi (line 63,"We... understanding"). Choice D is incorrect; while Akira may know that Chie is unaware of his feelings for Naomi, this is not what he is referring to when he mentions "a matter of urgency."