How easily you can get away with dropping a walk entirely depends on the dog and other contextual factors, size, health, age and temperament of dog, availability of alternative exercise and enrichment (such as garden), etc.
We luckily have a few places we can drive to where our dog can safely be let out of the car for 10 minutes while we sit in the car having a cup of tea, so in extremes we might do that. He hates the rain so 10 minutes is the maximum he would want if it is belting down. He actually is the sort of dog who can go a day without a walk on occasion if he has to, tho obviously we take him out if we can.
But your health MUST come first. No human should endanger themselves for their dog, firstly because humans are more important than dogs (and I fucking love dogs - but they just are), and secondly because your dog will suffer more in the long run if your illness is prolonged by not resting when you need to.
Anyone who thinks it is appropriate for a sick person to force themselves out in terrible weather, in order to avoid a dog feeling slightly bored and frustrated for one day, seriously needs to sort their priorities!
Ideally, if one person is ill, a friend or family member or even a paid dog walker might take the dog out (depending on current financial situation). But that isn't always possible.
In short, I would advise OP, and anyone, to do their best by their dog (assessing the dog's unique situation and needs) but to prioritise their own health and wellbeing.