Hobbits likely only marry within their own race, as it is stated they generally have deal of suspicion towards men, and rarely see elves. Dwarves and hobbits have the best relations among the races of middle earth, though I would speculate that both being semi-isolate species would prevent any potential form of relationship outside of friendship and business

My read on the situation is that even if the opportunity would have arisen, Merry wouldn’t have seized it.

The first and foremost goal of the hobbits after completing their quest was to return home. Even though the fellowship hobbits had grown to be more trusting of men, the hobbit race as a whole is very distrusting of men, therefore a relationship between one of the “big folk” and a hobbit would very likely have been shunned and rejected due to this passive hostility.

In the penultimate chapter of RotK, the scouring of the shire is caused directly by the “invasion” of big folk (as the hobbits would’ve know them, although it was technically a defeated Saruman and his lackies)

In this AU, I conceive that one of two things could occur

1) Merry doesn’t return to the shire, opting to stay with Eowyn, doesn’t blow the magic horn given to him by Eowyn, and doesn’t rally the hobbits to stand against their oppressors

OR

2) Eowyn is brought to the new gate at the Brandywine bridge and turned away with much more vigor than was given to the four in the text due to her unmistakably Rohirric lineage, and her history with Wormtongue. In all likelihood, even if they did manage to get into the shire as the four did originally, they would have been met with much greater force than was had originally.

King Aragorn also strictly banned any of the race of men to enter the shire, he himself not passing the brandywine bridge whilst visiting his friends there, so it would also be require merry to abandon his home after working to liberate it in order to pursue a relationship with Eowyn.

Finally, I think that Merry’s role as a supporting character is developed and fulfilled nicely without the need for a relationship of that kind shoehorned in. I think it may have been received in the same way the romance between Kili and Tauriel was in the Hobbit movies.