The chi-square test of homogeneity of proportions can be used to compare two sets of multinomial proportions over the same set of values. In that sense it can be thought of as a two-sample equivalent (or indeed more than two-) of the chi-squared goodness of fit test (in a similar sense to the way a two-sample Kolmogorov-Smirnov test relates to a one-sample test).
In short, yes, provided the conditions for the test to be suitable all hold to a sufficient-for-your-purposes approximation.
This procedure is basically the idea behind "CHi-squared Automated Interaction Detection", or "CHAID" described by G.V. Kass in 1980. The general setting is very similar to your television watching prediction example: You want to best predict the occurrence of a categorical variable by a combination of other categorical variables. You do this by finding the split with the maximal $\chi^2$ value.
A description of the algorithm and the issues around adjusting for statistical significance are given in (Kass, 1980). In that paper the Bonferroni correction is used to adjust for the selection of the maximal $\chi^2$ value.
Some actual theory is available for the case of reduction to a $2\times2$ table (Kass, 1975).
There is an R package called CHAID which implements the algorithm and is available on R-Forge.
Although it is a little different from your question, there is a similar situation that arises when dichotomizing a continuous variable to predict another dichotomous variable. Namely, where should you put the cut-point? This is discussed in (Miller and Siegmund, 1980) and (Halpern, 1982), among others.
Yet another setting where this type of question comes up is in change-point estimation or segmentation, though it has been too long since I looked at those papers to recall authors.
References:
Halpern, J. (1982). Maximally selected chi square statistics for small samples. Biometrics, 1017-1023.
Kass, G. V. (1975). Significance testing in automatic interaction detection (AID). Applied Statistics, 178-189.
Kass, G.V. (1980). An Exploratory Technique for Investigating Large Quantities of Categorical Data. Applied Statistics, 29(2), 119-127.
Miller, R. and Siegmund, D. (1980). Maximally Selected Chi-Squares. Technical Report 64. Stanford, Calif, Division of Biostatistics, Stanford University.
Best Answer
No test "approves" the null - you either reject it or fail to reject it. Failing to reject the null (as with your chi-square result) does NOT mean the two variables are independent, it means there is insufficient evidence of their dependence to reject the null
You don't give the CI or p for the correlation, but you state that it is negative and imply that it is significant. That does not contradict 1) because you are asking questions about different pairs of variables: One is categorical the other is continuous.
You state in a comment that you categorized the continuous variables in order to use chi-square. Binning a continuous variable is rarely correct. See here.
The first reason given at that link is a loss of power, thus, it would not be surprising to find the more powerful test having more significant results - that's more or less what power means.
You ask which you should keep. Clearly, I vote against the chi-square. But it is not necessarily the case that correlation is what you want, either. It depends on your question.