The answer is: the ISO-OSI stack.
OSI stands for Open Systems Interconnection (ISO is the International Standardization Organization), and it's a model which defines the structure used to transmit data between every kind of devices. Each level is a different abstraction layer, and adds rules or details that define the communication protocol.
While Internet (excluding things that are commonly associated with it, like HTTP) belongs to the upper layers (the Network layer), serial communication is just a way to define the Physical layer.
This is the OSI model stack, compared with the TCP/IP model used for Internet: you can see that Internet is defined at the network level, while the serial protocol (in strict sense, not the implementation) is defined by the physical layer, at the base of the stack.
From Wiki about the Internet protocol suite:
The Internet protocol suite is the set of communications protocols used for the Internet and similar networks, and generally the most popular protocol stack for wide area networks. It is commonly known as TCP/IP, because of its most important protocols: Transmission Control Protocol (TCP) and Internet Protocol (IP)
It has four abstraction layers, each with its own protocols. From lowest to highest, the layers are:
The link layer (commonly Ethernet) contains communication technologies for a local network.
The internet layer (IP) connects local networks, thus establishing internetworking.
The transport layer (TCP) handles host-to-host communication.
The application layer (for example HTTP) contains all protocols for specific data communications services on a process-to-process level (for example how a web browser communicates with a web server).
Ethernet and WiFi are examples of protocols that can work as Network Access Layer, providing the physical medium and the basic transmission rules (like the encoding of symbols) for the Internet connection.
Other protocols used at different layers of the stack are, as mentioned, TCP, UDP, HTTP and many others.