We study the spreading dynamics of content on networks. To do this, we use a model in which content spreads through a bounded-confidence mechanism. In a bounded-confidence model (BCM) of opinion dynamics, the agents of a network have continuous-valued opinions, which they adjust when they interact with agents whose opinions are sufficiently close to theirs. The employed content-spreading model introduces a twist into BCMs by using bounded confidence for the content spread itself. We define an analogue of the basic reproduction number from disease dynamics that we call an \emph{opinion reproduction number}. A critical value of the opinion reproduction number indicates whether or not there is an ``infodemic'' (i.e., a large content-spreading cascade) of content that reflects a particular opinion. By determining this critical value, one can determine whether or not an opinion dies off or propagates widely as a cascade in a population of agents. Using configuration-model networks, we quantify the size and shape of content dissemination by calculating a variety of summary statistics, and we illustrate how network structure and spreading-model parameters affect these statistics. We find that content spreads most widely when the agents have a large expected mean degree or a large receptiveness to content. When the spreading process slightly exceeds the infodemic threshold, there can be longer dissemination trees than when the expected mean degree or receptiveness are larger, even though the total number of content shares is smaller.