This paper first discusses the new mode of digital technology to boost the culture and tourism integration vacation industry and analyzes the specific application of digital technology in the culture and tourism integration vacation. Secondly, the index evaluation system is constructed with the goal of vacation destination competitiveness, and the gray time-lag prediction model is constructed by introducing the time-lag differential equation model combined with the gray correlation degree and the GM(1,1) model. Finally, an empirical analysis of competitiveness prediction was conducted to confirm the effectiveness of the gray time-lag prediction model. The results show that the tourism resource product has the highest comprehensive relevance, with a score of 0.9254, respectively. The use of the gray time lag prediction model can realize the prediction of the competitiveness score of the vacation destination and provide effective data support for the construction of a digital culture and tourism integration vacation system. This shows that the time-lag differential equation combined with the gray prediction model can be used to analyze the competitiveness score of the vacation destination, promote the deep integration of culture and tourism industry, and create a vacation system that better meets the expectations of tourists.