Hi Simran,
The short answer is no. You don't need a 32bit microcontroller to have an RTOS.
You are limiting your choices and capabilities though. I did a quick web search and came up with this thread:
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To summarize the thread, the following companies offer an RTOS for 8/16/32 bit architectures:
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Salvos RTOS
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Hope this helps you or someone else out.