If you can't put -1/-1 counters on Devoted Druid (due to an effect such as that of Solemnity), you can't activate its second ability. If you can put counters on it, but that is modified by an effect (such as that of Vizier of Remedies), you can activate the ability even if paying the cost causes no counters to be put on Devoted Druid.

You put the -1/-1 counter on Devoted Druid as a cost to activate its ability, not when it resolves. If paying the cost causes Devoted Druid to have 0 toughness, it's put into your graveyard before you can untap it and before you can even pay the cost again.