<p>I have a large CSV file and I want to execute a stored procedure for each line.</p>

<p>What is the best way to execute a stored procedure from PowerShell?</p>

Consider calling osql.exe (the command line tool for SQL Server) passing as parameter a text file written for each line with the call to the stored procedure.
SQL Server provides some assemblies that could be of use with the name SMO that have seamless integration with PowerShell. Here is an article on that.
http://www.databasejournal.com/features/mssql/article.php/3696731
There are API methods to execute stored procedures that I think are worth being investigated. Here a startup example:
http://www.eggheadcafe.com/software/aspnet/29974894/smo-running-a-stored-pro.aspx