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In brief: Sciatic pain. Yes. Ice is an excellent anti-inflammatory option. Try alternating moist heat to relax muscles and ice to reduce swelling. Physical therapy and stretching exercises are important additions to reduce the pain and prevent recurrence of the pain.
Just water: Ice is just water and unless you are talking about gallons every day, it has no real effect on blood viscosity. If you crave ice, though, it can be a sign of iron deficiency anemia so I would relay the fact you eat a lot of ice to your doctor next time you are in for an exam.