Quite at the beginning of Chapter 2 he explains:

"Matrimony was always one of my favourite topics, and I wrote several sermons to prove its happiness: but there was a peculiar tenet which I made a point of supportin; for I maintained with Whiston, that it was unlawful for a priest of the church of England, after the death of his first wife, to take a second, or to express it in one word, I valued myself upon being a strict monogamist."

So he certainly means that widowers shouldn't remarry, at least priests shouldn't.
Mr. Wilmot, however, isn't a priest (or am I missing something?) and so it should be all right for him.

In general I do not see the point of Primrose's argumentation. At a time when many women died in childbed, it was simply necessary for a widower to marry again, in order to have a woman to look after and to raise the children.

And besides, it is certainly not immoral to marry again if one really wants to, since this would have been the deceased's wish, too. From all we have learnt of the kind-hearted and loving Dr Primrose so far, I doubt he would want his wife be lonely and unhappy, once he was dead, only because she would not dare to remarry.