I managed to figure it out by looking at the discussion here: Elasticsearch EC2 setup across multiple regions. The solution is the last comment in the post.
In the elasticsearch.yml file for cluster 1, I added the following lines to the file:
network.publish_host: Cluster 1's public IP address
discovery.zen.ping.unicast.hosts: ["Cluster 2's public IP address"]
In the elasticsearch.yml file for cluster 2, I added the following lines of code:
network.publish_host: Cluster 2's public IP address
discovery.zen.ping.unicast.hosts: ["Cluster 1's public IP address"]
I then restarted elasticsearch and the clusters were able to see each other and connect to each other as remote clusters. From there I was able to cross cluster search on both clusters.