Had to have been made close to 1930's. As I remember it:
Opening:
Small town, idyllic, peaceful and beautiful street where passersby have scoffed at large statue of man on raring horse in the yard of elderly lady.
Fresh faced young woman decides to interview elderly lady to get the scoop on the statue.
Spends afternoon with matron who takes out an old sewing basket, removes aged baby keepsake while they settle and she begins story.
Fade to past when she is young and courted by dashing fellow.
He came to her on horseback (as feisty as he) and would throw stones at her window until she opened it.
Years pass, the story is not always happy. (Some if it over my head.)
Tragic scene when she gets into horse drawn carriage with baby basket of twins.
Galloping team, snow/ice and a bridge over a raging river, the carriage goes over into the water and the babies are swept away.
For some reason, the couple seems to face much adversity or his wild ways are always an issue.
But he was always her hero--thus the statue.
End:
Girl leaves with gratitude of story, walks by statue looking up with admiration/understanding--tears streaming down her face.

I asked my movie trivia son who says it doesn't ring a bell. I emailed a movie buff in NJ, so I might know something on Tuesday. Roger Ebert would know, but he's probably in bed by now (unless he is still at the party).

Did the man die in the river trying to save the babies?

That I can't remember for sure. I was young when I saw it on Sunday Afternoon at the Movies on TV in the 1960s. I watched those old movies every week for 12 years. Uncommon for a kid, huh.

My NJ contact emailed me that the summary you wrote doesn't ring a bell with him either, but he said the movie must have been a five-hanky one.

I will continue to do research. (Too bad libraries are closed today. I'm a librarian.)