No, Viewstate is coupled tightly with the use of the browser as an HTTP client.
For webservices, you have two choices: let the client track the state of the conversation, or let the server track it.
Using the server session state, and passing a cookie (either HTTP cookie or some cookie-like parameter inside the SOAP envelope)
requiring the client to track, retain, and possibly transmit to the server, the conversation state.
About Viewstate - it is state for the page as presented to the user, and as an implementation, it is tightly coupled to the browser. When the page is displayed, viewstate info is used to fill the page. Later, when a form on the page is posted, the relevant form data, some of which may have been preset with the viewstate magic, is then transmitted to the server. The server needs to validate the input from the client, despite the use of viewstate on the client side.
You can see that viewstate coupled with some lightweight browser-side population logic is a way for the client to manage the state of the page the user is seeing, but the server cannot forgo validation for conversation state.
This approach can be taken in a web services application, but because there is no dependence on the browser or on a particular presentation (or any presentation at all), it is a do-it-yourself thing. The client application maintains and uses any conversational state in a way that is appropriate for the client.
On the other hand, Server-managed state means there is state information retained at the server for each "conversation" or "session". The client doesn't necessarily need to track the information, if the server is doing it. The client just presents a token (or cookie if you like) to the server and the server uses it as a lookup key into a state table. The server is primarily responsible for validating all state retained on behalf of the client.
Since you're using .NET, you might be interested to learn that Workflow can be used server-side to track the state of a webservices (WCF) based conversation. This approach maintains the WS network protocols - it does not stipulate any particular client technology or platform.