Although I'm not a lawyer, I have tried to research this in relation to the UK. Other European countries may be different. I'd be interested if anyone knows if any of the following is incorrect.
If a default judgement is issued by a US court it is not directly enforceable in the UK. The judgement may be a monetary award and/or an injunction preventing further sale of the infringing product in the US.
For the UK court to recognise that the US court has jurisdiction over the defendant one of the following must be satisfied
The defendant has voluntarily entered into US court proceedings
The defendant was present or had a place of business in the US at the
start of proceedings
The defendant has submitted to the jurisdiction of the US court in a
The UK cases Cape vs Adams  and Lucas Film vs Ainsworth  are the best examples of this being tested. In both cases the court ruled that the US did not have jurisdiction over the defendant as he / it was not present in the US as the start of court proceedings. In Lucas Film vs Ainsworth it was established that selling to US customers is not a 'presence in the US'.
So, for the UK at least, if none of the above conditions are met, a US judgement cannot be enforced.
Whether the plaintiff would then start sending cease and desist letters to your web hosts and payment providers is another matter.
The question discussed 'continuing patent litigation' which implies that US proceedings have already been entered into.
It seems to me that UK companies with no presence in the US should think very carefully whether it's worth entering into US proceedings.