No, the gypsies had nothing to do with the death of the girl. Sir Arthur Conan Doyle put the gypsies in the story to lead the reader astray. In fact, it led Sherlock Holmes astray too. The gypsies were the only friends that Dr. Roylett had. He gave these wandering "bands" permission to camp on his property, and, in turn, they would invite him into their tents and allow him to wander with them for weeks at a time.
Helen Stoner tells Sherlock that when her sister died, she mentioned the speckled band. Sherlock asks her what she thinks that might mean. Helen replies,
" Sometimes I have thought that it was merely the wild talk of delirium, sometimes that it may have referred to some band of people, perhaps to these very gypsies in the plantation. I do not know whether the spotted handkerchiefs which so many of them wear over their heads might have suggested the strange adjective which she used." (page 8)
However, after Sherlock solves the mystery, he thinks that the idea of gypsies distracted him from the truth. Sherlock explains this to Dr. Watson as they return to London the following morning.
"I had," said he, "come to an entirely erroneous conclusion which shows, my dear Watson, how dangerous it always is to reason from insufficient data. The presence of the gypsies, and the use of the word "band" which was used by the poor girl, no doubt to explain the appearance which she had caught a hurried glimpse of by the light of the match, were sufficienit to put upon me an entirely wrong scent. I can only claim the merit that I instantly reconsidered my position when, however, it became clear to me that whatever danger threatened an occupant of the room could not come either from the window or the door." (pg 20)
So, initially, Sherlock felt that the gypsies were a possibility, but when he realized that no one could enter the room through the window or the door, then he knew that the threat was from inside the house. The only other entrance into the room was through the vent connecting Helen's room with Dr. Roylett's.