Assuming that no dog officially classed as 'dangerous' is involved, and that the keeper of the aggressive dog doesn't deliberately set it onto the other one, then no criminal offence is involved.
However Section 2 of the Animals Act 1971 means that the keeper of the aggressive dog (or the owner if the keeper is under 16) might well be held liable to pay for the 'damage' done by his/her dog. (i.e. a court could rule that he/she must pay the other party's vet's bills).
To win a claim for damages, the other party would have to show that the keeper (or owner) of the aggressive dog knew that his/her dog was likely to cause serious damage OR, that if damage was to occur (even though it might not have been regarded as likely), then such damage would be severe.
http://www.legislation.gov.uk/ukpga/1971/22