Could the rash on my baby's armpits be due to me picking him up?

Probably Not. Skin to skin contact usually doesn't cause a rash, so it is doubtful that picking up your child would cause a rash. Babies occasionally can have fungal infections under their armpits, given the moist environment. If your baby has a rash under their armpits and it is not improving, you should consult your physician.
Unlikely. Baby armpit rashes are usually due to excess moisture and/or some yeast overgrowth. Baby powder can be used in the armpits to make them drier. Be sure the baby doesn't inhale the powder. If the rash persists, a doctor should examine it to see if anti-yeast medicine is needed.
It's possible. If the rash occurs in the same locations each day after you pick him up it is likely that his skin is sensitive and your handling may be the cause of the rash. Check the ingredients of any hand lotions you may be using as a moisturizer. Some infants have very sensitive skin and react to fragrances and/or other ingredients. If the rash worsens, spreads or has blisters it may be an infection.