Below are the weekly Nielsen ratings results for the 1977-78 television season. Viewership wasn't reported by Nielsen until the fall 1988 season, so up until that point the only metric widely available was the household rating (which is what is reported here in this article). To calculate the estimated number of households watching, just multiply each rating by 729,000 as that is what 1 rating point represents.
Full weekly household ratings for every show weren't made available until 1981, so only partial results are available for this season. The first week of the season actually does have full results available, but the rest of the weeks usually only have ratings information available for the top 10 programs (from The Associated Press). Beginning with the w/e May 21, 1978 and through w/e Aug. 6, 1978 there are the top 15 results available (from from The New York Times). Unfortunately, there is no ratings data available for the w/e Dec. 18, 1977 or Dec. 25, 1977 - only rankings.
Additionally, the second document includes full rankings (without ratings) from both The Miami News and The Fort Worth Star-Telegram and these are available for most weeks of the proper season (not the summer) except for the following: week ending Oct. 16, 1977 (only top 10), Oct. 30, 1977, Dec. 25, 1977, Jan. 1, 1978, Jan. 8, 1978, and Jan. 15, 1978 (all only top 20). The final week of full rankings is w/e May 28, 1978.
ARCHIVED WEEKS (ratings - 51/53 weeks)
ARCHIVED WEEKS (full rankings - thru May 28, 1978)
Source: Nielsen Media Research via The Associated Press, The New York Times, The Fort Worth Star-Telegram, The Miami News, and Broadcasting Magazine