AWS employee here on the ECS team. ECS on Fargate would be the closest thing to what you are asking for. Upload a container image, create a Fargate cluster, and launch a service on that cluster that runs your container.
It's currently available in the us-east-1 region, under the ECS service in the console. Create a new cluster and Fargate will be an option for launching and operating the cluster.
It's frustrating how hit or miss their service availability is in each region. I can understand other countries with different laws and regulations but they can't even get some services multi-region in the US.