Does anyone know a no-arbitrage proof that call option delta has to be between 0 and 1? We know that Black Scholes model N(d1) tells us that it is between 0 and 1, but is there a no-arbitrage argument?
Depends what you mean by 'delta'. If you mean N(d1), then (as you say) it is by definition between 0 and 1. Or if you mean 'the sensitivity of option price to change in the underlying price' then you have to ask what is meant by 'option price'. If it means the price given by the standard BS formula, then again it comes down to N(d1). The price given by the standard BS formula is of course established using the no-arb assumption, so there is such a proof as you mention.
If you mean 'the sensitivity of observed price to change in observed underlying price', well that could be anything really. It's likely to be close to the theoretical delta for no-arbitrage reasons, but that's all.