BERLIN — A day after voters in Chancellor Angela Merkel’s political home state delivered what amounted to a strong rejection of her refugee policy, the German leader acknowledged on Monday that she was “very dissatisfied” with the result but insisted that she would stick with her chosen course.

Ms. Merkel, who was attending a summit meeting in China, waited 18 hours before addressing the unprecedented third-place finish of her center-right party in state elections on Sunday in the impoverished northeastern state of Mecklenburg-Vorpommern, where she has her own parliamentary constituency.

The nationalist, anti-migrant Alternative for Germany finished in second place, with 21 percent of the vote, behind the center-left Social Democrats, with almost 31 percent. Ms. Merkel’s Christian Democrats received 19 percent.

Ms. Merkel’s bloc of Christian Democrats and their Bavarian sister party, the Christian Social Union, had never been overtaken on the right in any state or national election in the history of modern Germany, and the result prompted the chancellor’s critics from her own camp and across the spectrum to renew speculation about her future.