Yes, but the connection is more linguistic than practical. 'Gothic' as a term originated as a pejorative description of a supposedly barbarous artistic style. 'Gothic' is one of the words art historians attach to a certain style in order to better classify and compartmentalise.
The Gothic style itself was a product of the medieval period, hundreds of years after the time of the Visigoths and Ostrogoths. It was not directly created or continued by the Gothic tribes, but rather developed from 'Romanesque' architecture, a style favoured particularly by the Normans which combined Ancient Roman and Byzantine forms with local traditions.
During the Renaissance, architectural interest in the buildings of Antiquity flourished. The styles that replaced this classical style were therefore naturally seen as corruptive and poor, when compared to the glories of the Roman style. Giorgio Vasari (d. 1574), named by some as the first 'art historian', used 'barbarous German style' to describe the artistic style now known as Gothic. He claimed that the various architectural features of Ancient Rome had been destroyed and mutilated by the Gothic tribes and replaced with a new style, which he called 'the Gothic'.
Therefore, the labelling of 'Gothic Architecture' has far more to do with perceptions of the Goths as destroyers of classical achievement, than it does with the Goths themselves actually developing it. John Evelyn, the 17th century English diarist, noted in his Account of Architects and Architecture that the best buildings are designed in accordance with Greek and Roman precedence. Westminster Abbey, amongst other English buildings, do not conform to these classical rules, and are written off as 'Gothick barbarism'.
Just as Renaissance thinkers found it necessary to portray medieval intellectual culture as one of ignorance and superstition, architectural writers found it necessary to criticise medieval architecture as barbarous and crude. From this supposed decline, the revival of classical styles could be seen as a good and moral thing to do. In order to criticise medieval architecture, it was linked to the tribes that sacked Rome- the worst architectural crime imaginable!
From a 19th century correspondent:
Authorities such as Christopher Wren lent their aid in deprecating the old medieval style, which they termed Gothic, as synonymous with everything that was barbarous and rude.
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