I don't think that this number will give you anything useful, but I found it a interesting problem to spend half an hour writing a little script to calculate.
I used the table from this Wikipedia page, which goes back to 1828 (the first election the popular vote was recorded for).
I also excluded the Republicans from the years that the table is using their column for another (presumably related) party. It doesn't make much difference, given how much larger the country is now.
The Republicans have been in 41 elections since 1856, with 24 wins, 10,352 total electoral votes, and 1,000,736,782 total popular votes.
The Democrats have been in 48 elections since 1828, with 22 wins, 10,161 total electoral votes, and 990,385,979 total popular votes.
It may be worth noting that 29.6% of the Republican popular votes and 31.4% of the Democratic popular votes have come in the 21st century (the 2000 election and on)