Sql-server – Same tables with same data, different SQL Server stored procedure performance

Yes it will be different. Every version will have tweaks to the optimiser, or code path, or caching algorithms or something. I would expect the later versions to run faster than the earlier ones.

I am using indexing to speed up the performance.

No, you're not. If every row in the table has to be updated then an index is, at best, irrelevant. If the index contains columns which you are updating then the index's pages also have to be updated incurring additional write overhead.

Have you checked the query plan for the three different instances? My guess is the 2008R2 has less memory available and the optimiser's choosing to spill to TempDB, or the operating system's forcing paging to the virtual memory pagefile. Or it may be that the transaction log files for the 8R2 instance started out small and what you're seeing is an artifact of autogrowth to accommodaate six million log records.

I would always recommend batching to my devs when processing this many records. Update the first N records in cluster key sequence. Then do the next N, and so forth until all 2M are done. "N" can be altered to give the best rows/minute on your particular installation, though the tens of thousands tends to be about right. In this way log files remain smaller, transactional locking is reduced and it is restartable in the case of a failure.

Assign the value of parameter to the local variable in the procedure then used that variable.