With two exceptions, the answer is no. Sea ice forms from seawater. Sea ice forms from early autumn to early spring but then melts from early spring to early autumn. If there is any change in volume due to sea ice being nearly pure water while seawater is salty (note well: this is very small change), it repeats.
One exception is addition of ice and liquid water to the oceans from glaciers sliding into the ocean. Once that glacial ice detaches from the glacier it becomes sea ice. However, the sea level rise occurs when the glacial ice slides into the ocean and detaches from the ground, rather than later when that new sea ice melts.
The other exception is that water expands when it gets warmer. Less sea ice can form as the oceans get ever warmer. It is not however the freezing and melting of sea ice that make sea levels rise. That said, the reduced amounts of sea ice is one of many signs that the oceans are warming, and those warming ocean temperatures contribute to sea level rise.
Until the start of this century, thermal expansion and meltwaters appeared to be coequal in terms of causing sea level rise. Since then, meltwaters from glaciers and land-based ice sheets have been contributing considerably more to sea level rise than thermal expansion.